GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
MOLECULAR (GRAPH) CHARACTERISTICS OF SOME
HYDROCARBONS THROUGH GRAPH THEORY
B. K. Mishra
Centre of Studies in Surface Science and Technology,
Department of Chemistry,
Sambalpur University,
Jyoti Vihar – 768 019, India
ABSTRACT
An organic molecule can be represented by a graph, which can be converted to
several matrices by using various graph characteristics. Connectivity of atoms
through bonds leads to adjacency and distance matrices. The polynomials,
generated from these matrices may be treated as the signature of those molecules.
The eigen values of these polynomials are also treated as molecular descriptors and
have been used in quantitative structure property/activity relationships. Similarly
the relationship of polynomials of a molecule with those of the synthon components
of the molecule leads to in silico synthesis.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
MOLECULAR
CHARACTERISTICS
2
OF
SOME
HYDROCARBONS THROUGH GRAPH THEORY
1. GRAPH THEORY AND CHEMISTRY
1.1. Introduction
To bring the power of mathematics to bear on real-world problems, the problem
should be first modeled mathematically. Graphs, representatives of mathematics, are
remarkable versatile tools for modeling. A graph, G (V,E), can be defined as a
mathematical structure consisting of a vertex set(V) and an edge set (E). Each edge has a
set of one or two vertices (also termed as nodes) associated to it. Graph Theory (GT) is
largely applied to the characterization of chemical structures, as well as to qualitative and
quantitative structure-property (QSPR) and structure-activity (QSAR) relations by means
of certain numerical characteristics, the topological indices.
Chemical graph theory is a branch of graph theory that is concerned with
analyses of all consequences of connectivity in a chemical graph. Chemical graph serves
as a convenient model for any real or abstracted chemical system1,2. It can represent
different chemical objects as molecules, reactions, crystals, polymers, clusters etc. The
common feature of chemical systems is the presence of sites and connections between
them. Sites may be atoms, electrons, molecules, molecular fragments, groups of atoms,
intermediates, orbitals etc. The connections between sites may represent bonds, bonded
and non-bonded interactions, elementary reaction steps, rearrangements, Van der Waals
force etc. Chemical systems may be depicted by chemical graphs using a simple
conversion rule:
Site ↔ vertex
Connection ↔ edge
Chemical graph theory (CGT) appears to be one of the most misunderstood areas
of theoretical chemistry. Randić3 tried to outline briefly causes for misunderstanding and
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
3
suggested remedies, including the test on the knowledge of GT and CGT. CGT should be
viewed not only as equal to other branches of theoretical chemistry but also as
complementary and necessary for better understanding of the nature of the chemical
structure.
The language of graph theory is different from that of chemistry. Therefore, a
graph theoretical terminology4,5,6,7 (table 1) is proposed for standard use in chemistry and
for the corresponding chemical terms.
Table-1: Mapping of graph theoretical and chemical terms.
GRAPH THEORETICAL TERMS
CHEMICAL TERMS
Chemical (molecular) graph
Structural formula
Vertex
Atom
Weighted vertex
Atom of a specified element (Mostly other
than carbon)
Edge (line)
Chemical bond
Weighted edge
Chemical bond between the specified
Elements
Degree of a vertex
Valency of an atom
Tree graph
Acyclic structure
Chain
Linear alkane or polyene
Cycle
Cycloalkane or annulene
Bipartite (bichromatic) graph
Alternant chemical structure
Nonbipartite graph
Nonalternant chemical structure
1-Factor (Kekule graph)
Kekule structure
Adjacency matrix (A)
Huckel(topological) matrix
Characteristic polynomial
Secular polynomial
Eigenvalue of A
Eigenvalue of Huckel matrix
Eigen factor of A
Huckel(topological) molecular orbital
Positive eigenvalue
Bonding energy level
Zero eigenvalue
Nonbonding energy level
Negative eigenvalue
Antibonding energy level
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
Graph-Spectral Theory
Huckel theory
Graph spectrum
Huckel polynomial equation
4
Molecular graphs are a special type of chemical graphs, which represent the
constitution of molecules1,8,9. They are also called constitutional graphs.10,11 When the
constitutional graph of a molecule is represented in a two-dimensional basis it is called
structural graph. Pogliani12 introduced the complete graphs for the inner core electrons.
Konstantinova and coworkers13 reported on the application of information theory
to the problem of characterizing molecular structures. The information indices based on
the distance in a graph are considered with respect to their correlating ability and
discriminating power.
Topological indices are structural invariants based on modeling of chemical
structures by molecular graph. Predih and Predih14 reported that the atom contribution
approach, the bond contribution approach, the contribution of terminal or interior atoms
or bonds approaches do not seem to be viable to perform the structural interpretation of
topological indices and physiochemical properties. On the other hand, the approach of
structural interpretation using the structural features like the size of the molecule, the
number of branches, the position of branches, the separation between them, the type of
branches, and the type of branched structure is generally applicable. Perdih15 derived
branching indices of the BIA groups. Gutman and coworkers16 derived relations between
topological indices of large chemical trees. Turker17 starting with the concept of T (A)
graphs for alternant hydrocarbons defined a novel topological index, which differentiates
isomeric as well as isospectral molecules very well. Torrens18 derived a new chemical
index inspired by biological plastic evolution.
1.2. APPLICATIONS OF CHEMICAL GRAPH THEORY
The expensive and time consuming process of drug lead discovery is significantly
accelerated by efficiently screening molecular libraries with a high structural diversity
and selecting subsets of molecules according to their similarity towards specific
collections of active compounds. To characterize the molecular similarity/diversity or to
quantify the drug-like character of compounds the process of screening virtual and
synthetic combinatorial libraries uses various classes of structural descriptors, such as
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
5
structure keys, finger prints, graph invariants and various topological indices computed
from atomic connectivities or graph distances. An efficient algorithm for the computation
of several distance based topological indices of a molecular graph from the distance
invariants of its sub-graphs is reported by Klein and Ivancius.19 The procedure utilizes
vertex- and edge- weighted molecular graphs representing organic compounds containing
heteroatoms and multiple bonds. These equations offer an effective way to construct
weighted molecular graphs and consequently to compute the Wiener index, even/odd
wiener index and resistance distance index. The proposed algorithms are especially
efficient in computing distance-based structural descriptors in combinatorial libraries
without actually generating the compounds, because any distance-based indices of the
building blocks are needed to generate the topological indices of any compounds
assembled from the building blocks. Wiener-type indices have been successfully used in
quantitative structure activity/property studies20, 21, 22, 23.
The symmetry group of a nonrigid molecule is related to that of the transition
structure that is related to the rearrangement process, which contributes to the
“nonrigidity” of the molecular system. The resulting permutation/ rotation/ reflection
groups for nonrigid molecules can be much larger in order, than the usual LonguetHuggins24 permutation/inversion group. By the group theoretical approach Bytautas and
Klein25 defined the symmetry group for nonrigid molecules.
A qualitative resonance-theoretic26 view is presented for the description of a
variety of conjugated π-network species identified with “sub-graphs” of a graphite
network. Within the framework of this resonance theory, simple rules are described to
provide qualitative information on ground state spin multiplicities; on patterns of groundstate spin density; and on exchange splitting to low lying “spin-flipped” excited states.
Beyond ordinary benzenoid molecules, illustrative applications are noted to be a diversity
of extended species, including differently structured edges on semi-infinite graphite;
corner structures (where edges along different directions meet); conjugated polymer-strip
ends; and local defect vacancy structures in extended graphite. The varieties of simple
resonance-theoretic predictions are compared against a semi-empirical unrestricted
Hartee-Fock view of some quantitative tight binding molecular-orbital computations.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
6
Agreement in predictions from the resonance- and band-theoretic viewpoint is taken to
engender reliability of the predictions. A traditional organic chemical resonance-theoretic
view is thence conveniently reformulated and brought to bear on several extended nanostructured systems to reveal systematic patterns of π-electronic behaviour.
The effects of different types of boundaries on graphite fragments are considered
as they influence the π-electrons. From a simple resonance theoretic argument there are
proposed simple structural conditions governing the occurrence of “unpaired” π-electron
density near the edges.27 Predictions based on these rules are made for a variety of edge
structures. Further, the novel resonance theoretic argument and predictions are
strengthened through more elaborate considerations of both the valence bond and
molecular orbital theoretic nature, especially for translationally symmetric polymer strips
with various types of edges.
1.3. GRAPH THEORETICAL DESCRIPTORS IN QSAR/QSPR STUDIES
Molecular graph is an important tool in drugs designing. Descriptors can be
engendered from molecular graphs through construction of matrices and graph
enumeration. Estrada and coworkers28 made a review on the use of topological indices in
drug design and discovery. Natarajan and Nirdosh29 worked on the application of
topological indices to QSAR modeling and selection of mineral collectors. Duchowicz30
and coworkers modeled the free-energy of hydrocarbons on the basis of topological
indices defined from the distance and detour matrices with in the realm of the QSPR
theory. Sharma et.al.31 calculated the excess isentropic compressibility, KsE values
employing density values of the binary and ternary mixtures and graph theory. Basak et
al.32 employ topostructural (TS) and topochemical (TC) indices, geometrical descriptors
and ab initio quantum chemical indices either alone or hierarchically in the development
of QSAR models of the aryl hydrocarbon (Ah) receptor binding potency of a set of 34
dibenzofurans.
Gupta et.al.33 investigate the eccentric-adjacency topochemical indices to estimate
anti-HIV activity of 107 derivatives of 6-(phenylthio) thymines. Agrawal and
coworkers34 used the distance based topological indices in modeling antihypertensive
activity of 2-aryl-imino-imidazolidines. Khadikar and coworkers35 use equalized
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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electronegativity (χeq) in modeling toxicity of nitrobenzene. Results show that more
reliable models can be obtained when χeq is combined with topological indices. Mishra
and coworkers have used graph theoretic parameters for correlating various physico
chemical parameters of alkanes36-42 ethers43,44 and some solvents45; critical micelle
concentration of some nonionic surfactants46; and mutagenic activities of amino acids47.
1.4. MATRIX REPRESENTATION OF STRUCTURAL GRAPHS
For the purpose of reflecting molecular topology and correlating structure and
properties quantitatively, graphs are converted in to a mathematical expression, which
may be a matrix, a polynomial, a sequence of numbers or a numerical index. The first
representation of a molecule using a matrix is due to Huckel matrix, which is used to
derive mathematical expressions of an organic molecule.
Some important and widely used matrix representations for characterizing a graph
are discussed below.
1.4.1. Adjacency matrix
The vertex-adjacency matrix A (G) of a labeled connected graph G with N
vertices is a square N×N symmetric matrix, which contains information about the internal
connectivity of vertices in G. It is defined as,
(A)ij= 1, if vertices Vi and Vj are adjacent
0, otherwise
(1)
(A)ii = 0
(2)
Adjacency matrix finds wide applications in the fields of chemistry and physics48-52
Example of adjacency matrix:
1
2
4
3
Hydrogen depleted structural graph of methyl cyclopropane
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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The vertex-adjacency matrix of G is
0 1 0 0
A(G)=
1 0 1 1
0 1 0 1
0 1 1 0
The adjacency matrix is symmetrical about the principal diagonal. Therefore the
transpose of adjacency matrix A leaves the adjacency matrix unchanged, i.e.
AT (G) = A (G)
(3)
The edge-adjacency matrix (EA(G)) of a graph G is defined as
(EA )ij = 1, if edges ei and ej are adjacent
0, otherwise
(4)
(EA )ii= 0
(5)
Example of edge adjacency matrix
For the above graph G, the edge-adjacency matrix is
0 1 0 1
EA(G) =
1 0 1 1
0 1 0 1
1 1 1 0
The vertex- and edge-weighted graph53-60, GVEW, is a graph, of which one or more
vertices and edges are distinguished in some way from the rest of vertices and edges. For
adjacency matrix of GVEW the equation (1) and (2) should be modified61-63 as
(A )ij
=1, if vertices vi and vj are adjacent and if edge (vi, vj) is K-weighted
=0, otherwise
(Aij)=h, if there is a loop of the weight h at vertex (i) in GVEW
(6)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
= 0, otherwise
9
(7)
Example
3
4
5
k
k
2
1
Hydrogen depleted structural graph of tetrahydropyrrole
The adjacency matrix of the above graph is
h k 0 0 k
k
0 1 0 0
A(GVEW)= 0 1 0 1 0
0 0 1 0 1
k 0 0 1 0
The Mobius systems are defined as cyclic arrays of orbitals with one or
more generally, with an odd number of phase delocalization resulting from negative
overlap between the adjacent 2PZ- orbitals of different signs64. Mobius system can be
depicted by Mobius graphs65-67 (GMO) which is defined as (V (GMO), E+(GMO), E- (GMO))
where, V (GMO)=vertex set
E+(GMO)=E+
Edge-subsets
E-(GMO)=EThe adjacency matrix for Mobius graph is defined as,
1, if vertices v and v are adjacent
(A)ij=
-1, if vertices v and v are adjacent and edge (vi,vj) is –1
weighted
0, otherwise
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
(A)ii=
0
10
(8)
Example
1
3
2
4
-1
Hydrogen depleted graph of cyclobutane
The vertex-adjacency matrix of the graph GMO is
A (G ) =
0
1
0
1
1
0
−1 0
0 −1
0
1
1
1
0
0
If a bipartite graph is labeled that 1,2-----, s are starred and s+1,s+2, ------, s+u unstarred
vertices then (A)=0 for 1≤ I, j ≤ s, s+1 ≤ I, j ≤ s+u.
Example
*
2
*
1
4
3
5
*
6
Hydrogen depleted graph of 1,2 dimethyl cyclobutane
The adjacency matrix of the above graph is,
0 0 0 1 0 0
0 0 0 1 1 0
A(GMO) =
0 0 0 1 1 1
1 1 1 0 0 0
0 1 1 0 0 0
0 0 1 0 0 0
A directed graph is called digraph. The adjacency matrix of a digraph AD(G) is defined as
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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AD(G)= the number of arcs from Vi to Vj if Vi≠Vj
and the number of self-loop at Vi if Vi= Vj
(9)
The digraphs can find wide applications in the vectorial electron flow in a system or in
representing chemical reactions.
1.4.2 Distance matrix
The distance matrix D = D (G) of a labeled connected graph G is a real symmetry
N×N matrix whose elements, dij are defined as7, 68,69,70
dij = l if i≠j
= 0 if i=j
(10)
Where, dij = length of the shortest path i.e. minimum number of between n vertices Vi
and Vj.
Example
1
2
4
5
3
Hydrogen depleted graph of cyclopropyl cyclopropane
The distance matrix D (G) of this graph is
0 1 1 2 3 3
D (G)=
1 0 1 1 2 2
1 1 0 2 3 3
2 1 2 0 1 1
3 2 3 1 0 1
3 2 3 1 1 0
The distance matrix of a labeled edge-weighted graph GEW is a real symmetric
N×N matrix whose elements dij are defined as71
dij = wij,
if i≠j
= 0,
if i=j
(11)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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Where, wij = minimum sum of weights of edges along the path between Vi and Vj
which not necessarily the shortest possible.
Example
1
4
2
5
3
Hydrogen depleted structural graph of tricyclo [2,1,0,03,5] pentane
0 1 3 2 2
1 0 2 3 1
D (GEW)= 3 2 0 2 1
2 3 2 0 3
2 1 1 3 0
The distance matrix D (GVEW) of a labeled vertex and edge-weighted graph GVEW is a
real-symmetric N×N matrix whose element, dij, is defined as72-74
dij = wii if i≠j
= wij if i=j
(12)
Where wii=weighted of a vertex V
wij=minimum sum of the edge-weights Kij along the path between the vertices Vi
and Vj which is not necessarily the shortest possible.
1.5. GRAPH MECHANICAL PARAMETERS
The intrinsic characteristics of a molecule can be determined from the quantum
mechanical
parameters
obtained
by
mathematical
operations
on
quantitative
characteristics of constituent atoms. The graphical representation of a molecule delineates
connectivity among the constituent atoms, which also provides information on their
interactions. Similar quantum mechanical operations can be applied for graph theoretical
interactions and various parameters such as, energy, bond order, electron density, graph
angle etc. can be obtained.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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1.6. Isomer Enumeration:
Isomers are chemical compounds with an identical molecular formula which
display at least some differing physical or chemical properties and which are stable for
periods of time that are long in comparison with those during which measurements of
their properties are made. Compounds with isomers are called unimers75. Traditionally
isomers have been classified as either structural isomers or stereoisomers76. Structural
isomers or constitutional isomers differ in there structures i.e. in the manner of bonding
the atoms in the molecules77. Stereoisomers have identical structures but they differ in
their configuration or conformation i.e. in the special architecture of the molecule.
The enumeration of isomeric structures is one of the oldest uses of the graph
theory in chemistry. There are a number of methods available for the isomer
enumeration.
1.6.1
The Cayley generating functions:
Cayley78 was first to attempt to enumerate the isomeric alkanes CH and alkyl
radicals CNHN+1. He represented the carbon skeletons of alkanes and alkyl radicals
by rooted trees in which the maximum vertex valency is four. These trees are also
called the Caley trees79.
Example
The graph theoretical representations of isomeric pentanes C5H12 by means of
trees are
and that of pentyl radicals C5H9 are ;
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
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1.6.2.Enumeration of Trees:
Cayley80 first used the name tree in 1857,although Kirchhoff81 first utilized the
concept in his fundamental work of electrical networks in 1847.Cayley developed a
generating function for enumeration of rooted trees,
(1 − X) − A o (1 − X 2 ) − A1 (1 − X 3 ) − A 2 (1 − X 4 ) − A3 − − − − − (1 − X N ) − A N −1
= A 0 + A1X + A 2 X 2 + A 3 X 3 + − − − − − − + A N X N
(13)
Where X=variable
N=number of vertices in the rooted tree
AN=number of rooted trees in a given N
A0=1 by definition
Ten years later, Jordan82 discovered the existence of the center and the bicenter of
the tree. Every tree has a center or a bicenter, but not both. Thus, tree with a center are
called centric trees and while those with a bicenter are called bicentric trees.
Example
(bicentric tree T1)
(centric tree T2)
Cayley made use of Jordan and enumerated the centric and bicentric tree. The
sum of centric and bicentric trees83 produced the total number of isomeric trees for a
given N.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
15
Later Caley84 improved his method by using the generating functions and already
known AN numbers for the enumeration of rooted trees. Thus, the trees TN can be counted
by means of the formulae:
T1=1
(14)
T2=(1/2) A1 (A1+1)
(15)
T3=the coefficient at X2 in (1-X)-A1
(16)
T4=(1/2) A2 (A2+1)+the coefficient at X3 in (1-X)-A1
(17)
T5=the coefficient at X4 in (1-X)-A1 (1-X2)-A2
(18)
And so on.
Cayley’s counting formulae have many errors. Half a century later Otter derived
an elegant formula for counting trees in terms of rooted trees,
T (X)=A (X)-(1/2)[A2 (X)-A (X2)]
(19)
Where A (X) is the counting series for the rooted trees.
The enumeration of trees by means of the Otter counting formula is carried out as:
A (X)=X+X2+2X3+4X4+9X5+20X6+48X7+---------
(20)
A2 (X)=X2+2X3+5X4+12X5+30X6+74X7+----------
(21)
A (X2)=X2+X4+2X6+4X8+----------------------------
(22)
(1/2)[A2 (X)-A (X2)]=X3+2X4+6X5+14X6+37X7+---
(23)
⇒T (X)=X+X2+X3+2X4+3X5+6X6+11X7+----------
(24)
The coefficients in the equation 12 represent the counts of trees with a given number of
vertices.
1.5.8.1.Methods for Enumeration of Alkanes:
Cayley82 first enumerated acyclic hydrocarbons by the application of
mathematical theory of trees. He enumerated alkanes and alkyl radicals up to 13
carbon atoms, but the number of isomers obtained for C12 and C13 alkanes, and
C13 alkyl radicals were incorrect. Almost immediately after the Cayley paper on
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
16
enumeration of alkanes, a work by Schiff86 appeared in which he correctly
counted distinct alkanes, alkenes and alkyl radicals with up to 10 carbon atoms.
Schiff also have the same error in counting for C12 as that of Cayley. The errors in
computing the number of C12 and C13 alkanes were first corrected by Herrmann
five years later. However, none of the above authors and several others was able
to produce a reliable method for enumeration of alkanes with large N. The first
significant advance after Cayley came in 1931 when Henze and Blair87-91, at the
university of Texas and Austin, developed recursion formulae for enumeration of
alkanes and related acyclic structures.
1.5.9. Periodic Table for Isomer Enumeration:
A formula “periodic table”92 for the isomer classes of all acyclic hydrocarbons
CnH2m has been proposed, with rows and columns respectively specifying numbers of C
atoms and half the H atoms. Asymptotic n→∞ behaviour of these enumerations is
developed, first for fixed degree u ≡ n + 1 - m of unsaturation and second for fixed
number 2m of H-atoms. The first-set isomer classes increase in size exponentially fast
with n, where as with the second set, the isomer-class sizes increase sub-exponentially, as
a power of n.
Beyond enumeration various properties of the different isomer classes may be
surmised to vary in a systematic manner with position in the periodic table. Such
systematic property variation is already generally quantitatively understood for the table’s
alkane diagonal sequence, which is studied in some quantitative detail in three other
works93-95. A similar study96 has been made for property variations of fully conjugated
acyclic polyenes. The isomer counts reported here has constitute a first step in such a
more extensive study of the whole of the current “periodic table” for acyclic
hydrocarbons, with the second step97 concerning distributions of graph invariants with in
an isomer class. A third step98 concerns similar distributions for “cluster-expansion”
approximants for selected molecular properties.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
17
1.5.10. Graph Cyclicity Index:
A new graph-theoretic cyclicity index99 C (G) is defined, being motivated in
terms of mathematical concepts the theory of electrical networks. This “global bond
excess conductance” index C (G) then is investigated, with a number of theorems as well
as some discussion and numerical investigation. It is found that C (G) typically has less
degeneracy than the standard cyclomatic number and has some intuitively appealing
features.
1.5.13. Graph Energy:
Let G be a graph possessing N vertices and let λ1,λ2, ----------,λN be its eigenvalues.
These eigenvalues will be labeled so that
λ1≥λ2≥ ------------ λN
If G represents the molecular graph of a conjugated hydrocarbon, then the total πelectron energy of this hydrocarbon in the HMO approximation is equal to Nα+Eβ100,101.
Here α andβ are the standard HMO-theoretical parameters where as, E is the quantity
depending on the eigenvalues of the graph G as
N/2
E=E (G)+2 ∑ λi
(25)
I =1
For the majority of the graphs first N/2 eigenvalues are positive valued and the
other half are negative valued.
Hence (25) can be written as
E (G)=2∑+λi
(26)
Where ∑+ indicates summation over positive eigenvalues.
On the other hand for all graphs,
N
∑ λi = 0
i =1
∴
Combining equations 26 and 27 one arrives at
(27)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
N
E (G)= ∑ |λi|
18
(28)
i =1
The quantity E in equation (28) is traditionally interpreted in terms of total πelectron energy. Almost all results obtained in the theory of total π-electron energy
assume the validity of equation (28). Therefore, some scientist call E (G) as a graph
invariant defined as the sum of absolute values of all eigenvalues of a general graph G
and named it “graph energy”. The definition of this novel graph invariant was first
published in 1978 in a mathematical journal that is not easy to find but later re-stated in
the book and elsewhere.
Markovic102 approximate the total π-electron energy of phenylenes by means of a
linear combination of spectral moments of both molecular and line graphs.
The idea to use E as a molecular structure descriptor, for predicting physicochemical properties of saturated organic molecules, seems to be first expressed by
Rakshit et.al103,
104
. Some ten years later Randic, Vračko and Novič105 put forward an
identical suggestion. They communicated a limited number of correlations that hinted
towards the possible applicability of E in QSPR/QSAR studies.
The total π-electron energy of an alternant hydrocarbon described by graph
G (N; e) is given by
E= 2 ne cos θ
(29)
Where n=N/2
=(N-1)/2, for odd alternant hydrocarbons. [∴N=number of vertices (atoms)]
e =number of edges (bonds)
and θ =angle of total π-electron energy.
θ is found to be very much useful in studying and encoding the fine topologies of
structural isomers of alternant hydrocarbons. In cases of isospectral structural isomers,
their θ values have to be same as according to the above equation (29). Then, obviously
their fine topologies have to be encoded not only by θ alone but some other topological
variants as well. One of them is azimuthal angle106 (φ). An angle φ could possibly exist
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
19
for any structural isomer of nonisospectral too but in the case of isospectral structural
isomers its role is great108.
Now, in a three-dimensional linear space vectors can be defined as C(0,0,√n) and
Di(xi,yi,z) such that norm Di=√e, tanθ =r/z
Where, r =√(xi2+yi2). Note that (C, Di)= √(ne)cosθ≡√nz.
The above treatment can be generalized including the odd alternant hydrocarbons
because these systems possess non-bonding molecular orbital having zero energy. The set
of vectors {Di}(generatrix) constitute a vector field (conical surface) in three dimensional
Euclideanspace109, 110, which has the form of an upside down cone shown in the Fig-1
whose apex is on the origin of the Cartesian
coordinate system (x, y, z), and its
height is z.
Z
Di
yi
Y
r
xi
X
Fig-1: Vector Field of a Compound
The angle between the vector Di and the z-axis represents the angle of graph
energy θ and the angle between r and xi represents the azimuthal angle φ where
φi=arctan(yi/xi). Hence each compound is characterized with four variables n, e, θ and φ.
Thus graph theoretical enumeration led to establish different characteristic
features of organic molecules. For the limitation of undertaking very large molecules in
quantum mechanical calculation, graph theoretical methods take the stage by using
fragmented but similar groups for various eigen-functions. In the present programme
small organic molecules containing six carbon atoms have been subjected to graph
theoretical manipulation for determining different eigen functions.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
20
2. CALCULATION OF GRAPH THEORETICAL PARAMETERS
2.1. Representation of molecules in matrix
For calculations all the possible structures of molecules containing up to 6 carbon
atoms and having maximum possible number of conjugated double bonds have been
considered. The number of possible structural isomers of C2 is one, which is an ethene
molecule. The number of possible structural isomers of C3 is 2 where one is cyclic and
other is acyclic. For C4 system the number of structural isomers are found to be 6 where 2
are acyclic and 4 are cyclic. For C5 system the number of structural isomers are found to
be 17 out of which 3 are acyclic and 14 are cyclic. From all the possible structural
isomers of C6 here 12 are considered.
All the structures of the molecules are given in table –1. The hydrogen depleted
structural graphs of the above graphs of the above molecules are considered for obtaining
matrices for further calculations. All the structural graphs of above 38 molecules are
given in the table-1.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
21
Table-1: Structural graph, adjacency matrix, polynomial and roots of some 38 molecules
Graph No.
Molecular structure
H
Structural Graph
H
1
2
1
1
2
H
1 x
x
2
H
3
H
2
1
1
H
H
3
x
H
1, -1
0, ±1.41421
x3-2x
3
2
2
1 1
1 x 1
1 1 x
1
3
x3-3x+2
-2,1,1
H
H
H
x 1 0 0
1 x 1 1
H
1
H
1
2
H
2
H
3
4
H
H
0 1 x
4
3
4
3
2
1
H
x -3x
x4-3x2+1
±0.61803
±1.618034
2
0 1 0 x
H
2
H
0
0, 0,
±1.73205
4
H
H
5
x2-1
1 0
1 x 1
0 1 x
1
4
Roots (x)
H
H
3
Polynomial
H
H
2
Adjacency matrix
x 1
4
H
H
1
3
x 1 0 0
1 x 1 0
0 1 x 1
0 0 1 x
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
H
H
x 1 0 1
3
2
3
2
6
4
1
H
1
7
2
4
1
x 1 0 0
2
1 x 1 1
4
3
0
x4-4x2
x4- 4x2+2x+1
0 1 x 1
3
-2.56155, 0,
1, 1.56155
x 1 1 1
1
4
1
4
2
3
1
4
2
3
1 x 1
0
x4-5x2+4x
1 1 x 1
2
3
-2.17009, 0.31111, 1, 1
0 1 1 x
H
H
8
±2, 0,0
1 0 1 x
2
H
1 x 1
0 1 x 1
4
H
1 CH
22
1 0 1 x
H
x 1 1 1
1
4
9
2
3
1 x 1 1
x4-6x2+8x+3
1 1 x 1
-3, 1, 1, 1
1 1 1 x
1
10
x 1 0 0 0
CH 2
1
2
HC
2
4 CH
2
5
CH 2
1 x 1 1 1
2
3
0 1 x
5
4
0 0
0 1 0 x
0
0 1 0 0 x
x5-4x3
-2, 0, 0, 0, 2
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
H
H
x 1 0 0 0
2
11
1 CH
3
4
2
4
2
23
5
CH 3
1
3
1 x 1
5
0 0
0 1 x 1
0
x5-4x3+3x
0, ±1,
±1.73205
0 0 1 x 1
H
0 0 0 1 x
H
12
4
CH
1
CH
2
x 1 0 0 0
2
2
4
3
5
CH
1
33
0 0
0 1 x 1 1
3
2
1 x 1
0 0 1 x 0
0 0 1 0 x
5
x 1 0 1
H
13
CH
4
5
3
1
1
2
H
H
1
H
4
1
2
5
3
5
3
x -5x +2x
x 1 0 0 1
H
2
0 0
0, ±2.13578,
±0.662153
0 0 0 1 x
H
H
14
3
2
H
1 x 1
0
0 1 x 1 0
1 0 1 x 1
4
3
x5-4x3+2x
±1.847759
±0.765367, 0
H
3
5
1 x 1
0 0
4
0 1 x 1 0
0 0 1 x 1
1 0 0 1 x
x5-5x3+5x+2
-2,
0.617999, 0.617999,
1.618035,
1.618035
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
H
H
1
x 1 0 0 1
1
5
2
15
24
2
5
3
4
1 x 1
0 1 x 1
4
3
3
2
x -6x +2x +4x
0
-2.48119, 0.68889, 0,
1.170087, 2
0 0 1 x 1
H
H
0 1
5
1 1 0 1 x
x 1 0 0 1
H
16
1
H
1
5
4
0
0 1 x 1 1
3
4
3
1 x 1 1
2
5
2
x5-7x3+4x2+2x
-2.85577, 0.32164, 0, 1,
1.7741
0 1 1 x 1
1 0 1 1 x
x 1 0 0 1
1
H
17
1
H
5
2
4
3
2
5
1 x 1 1 1
0 1 x 1 1
4
3
H
18
4
1
5
1
3
H
2
H
2
4
3
-0.35793, 3.3234, 1, 1,
1.68133
0 1 1 x 1
1 1 1 1 x
x 1
CH 3
x5-8x3+10x2-x-2
0 1
0
1 x 1 1
0
0 1 x 1 0
1 1 1 x 1
0 0 0 1 x
5
3
2
x -6x +4x +2x
0,
2.68554, 0.3349,
1.27133,
1.74912
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
25
x 1 0 0 0
H
19
5
4
3
4
3
2
1
CH
5
H
1 x 1
0 1
5
3
2
x -6x +4x +3x-2
0 1 x 1 1
2
1
0 0 1 x 1
2
0 1 1 1 x
x 1 0 0 0
H
20
4
5
1 x 1 1 1
2
5
1
CH
5
3
4
3
2
x 1 1 1
5
3
1
1
H
22
5
4
3
2
4
1
5
4
3
2
0
0 1
x5-7x3+4x3+2x
-2.85577, 0.32164, 0, 1,
2.17741
x5-7x3+6x2
-3, 0, 0, 1, 2
0
1 0 1 x 1
0 1 0 1 x
1
4
1 x 1
1 1 x 1
3
2
H
x5-7x3+8x2-2
-3.08613, 0.42801, 1, 1,
1.51414
0 1 1 x 1
0 1 1 1 x
3
H
2
0 1 x 1 1
1
H
21
-2.64119, 0.72374,
0.58922, 1,
1.77571
x 1 0 1
0
1 x 1 1 1
0 1 x 1 0
1 1 1 x 1
0 1 0 1 x
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
26
x 1 0 0 1
H
23
5
5
1 x 1
4
4
H
1
0 1 x 1 1
1
2
3
2
3
0 1
x5-7x3+6x2+3x-2
0 0 1 x 1
1 1 1 1 x
1
H
24
CH
1
2
1 x 1
2
3
3
H
4
0 0
0 1 x 1 1
4
H
x 1 0 0 0
H
25
1
4
2
5
CH
CH 2
H
26
3
H
3
H
1
2
H
5
2
1
3
4
5
0 1 x 1 1
5
x 1 1
0 0
1 x 1
0 0
1 1 x 1 1
0 0 1 x 1
0 0 1 1 x
H
x5-5x3+2x2+3x
-2.30278, 0.61803, 0,
1.30278,
1.61803
0 1 1 x 1
0 0 0 1 x
1
4
1 x 1 1 1
4
2
2
3
x5-5x3+2x2+4x-2
2.21432, -1,
0.53918, 1,
1.67513
0 0 1 x 1
0 0 1 1 x
5
5
H
x 1 0 0 0
2
2.93543, 0.61803,
0.4626,
1.47283,
1.61803
x5-6x3+4x2-5x-4
2.56155, -1,
1, 1, 1.56155
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
27
x 1 0 0 0 0
27
1 CH
2
H
1
2
4
CH
CH 3
5
3
6
3
1 0 0 0
0 1 x 1 1 1
2
3
CH 3
6
1 x
0 0 1 x
4
6
4
x -5x +3x
2
0, 0, ±0.835,
±2.07431
0 0
0 0 1 0 x 0
0 0 1 0 0 x
5
x 1 0 0 0 0
28
4 CH
CH
1 2
2
1
3
2
6 CH 3
5
6
1 0 0 1
0 1 x 1 1 0
0 0 1 x 0 0
3
2
CH
3
5
1 x
4
0 0 1 0 x
x6-5x4+4x2
0, 0, ±1, ±2
0
0 1 0 0 0 x
x 1 0 0 0 0
5
29
5 CH
4
H
3
4
6
3
6
CH 3
2
H
CH
1 2
1 x 1 0 0 0
0 1 x 1 0 0
3
2
1
0 0 1 x 1 1
0 0 0 1 x
0 0 0 1
0
0 x
6
4
x -5x +5x
2
0, 0,
±1.17554,
±1.90211
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
28
x 1 0 0 0 0
H
H
30
1 x
3
3
1
CH
5
2
4
CH
2
6
CH
2
1
2
4
5
1 0 0 0
0 1 x 1
0 1
0 0 1 x 1
6
x6-5x4+5x2-1
0
0 0 0 1 x 0
0 0 1 0 0 x
2
x 1 0 0 0 0
H
31
CH
2
CH
3
2
1
±0.51764,
±1, ±1.93185
6
4
5
H
H
2
1
1 x
4
3
5
1 0 0 0
0 1 x 1 0 0
0 0 1 x 1 0
x6-5x4+6x2-1
±0.44504,
±1.24698,
±1.80194
0 0 0 1 x 1
0 0 1 0 0 x
x 1 0 0 0 0
32
H
5
6
1
H
CH
2
H
4
4
5
6
3
1
2
3
1 x 1 0 0 0
0 1 x 1 0 1
2
0 0 1 x 1
0
0 0 0 1 x 1
H
0 0 1 0 1 x
x6-6x4+6x2
0, 0,
±1.12603,
±2.17533
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
29
x 1 0 0 0 1
H
H
33
5
4
5
1
2
H
1
1 x
1 0 0 0
3
0 1 x 1
2
0 0 1 x 1
6
3
6
4
0 1
x6-7x4+7x2-1
0
0 0 0 1 x 1
1 0 1 0 1 x
H
x 1 0 0 0 0
1 CH
34
1
2
2
6
H
H
1 x
2
6
3
3
5
5
4
H
2
H
2
6
3
3
5
x6-6x4+9x2-4
±1, ±1, ±2
x 1 0 0 0 1
6
H
0 1 x 1 0 0
0 0 1 x 1 0
0 1 0 0 1 x
1
1
H
x6-6x4+8x2-1
±0.37309,
±1.32132,
±2.02852
1 0 0 1
0 0 0 1 x 1
4
H
H
35
±1,
±0.41421,
±2.41421
H
4
H
5
4
1 x 1 0 0 0
0 1 x 1 0 0
0 0 1 x 1
0
0 0 0 1 x 1
1 0 0 0 1 x
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
30
x 1 0 0 0 1
H
1
36
4
5
H
3
6
3
4
5
2
3
4
5
4
5
0 1 x 1
0 1
x6-8x4+2x3+10x2+2x-1
0
1 1
0 0
0 1 x 1 0 1
0 1 1 x 1 0
-2.74108,
-0.71029,
-0.61803,
0.23136,
1.61803,
2.22001
-3, -1, 2, 2, 0,
0
x 1 0 0 1 1
1 x
3
6
6
0 0
0 0 0 1 x 1
1 0 1 0 1 x
2
37
1 1
0 1 1 x 1
1
1
1 x
2
2
1
x6-9x4+4x3+12x2
1 0 0 1 x 1
1 0 1 0 1 x
H
2
1
38
3
6
2
3
4
H
x 1 1 0 0 0
1
4
5
6
1 x 1 0 1 0
1 1 x 1 0 0
0 0 1 x 1 1
5
0 1 0 1 x 1
0 0 0 1 1 x
x6-8x4+4x3+12x2-8x
-2.73205, 1.41421, 0, 2,
1.414212,
0.73205
For converting the graph into a matrix here we have considered only the adjacency
matrices of the molecules. The adjacency matrix (A) can be defined as,
Aij=1, if vertices Vi and Vj are adjacent
=x, if vertices Vi=Vj
(30)
=0, otherwise
For example for molecule (I)
H
2
3
1 CH 2
HC
3
(I)
2
the structural graph is
x
represented as
3
1
1
and hence its adjacency matrix by definition (1) can be
0
1 x 1
0 1 x
All the adjacency matrices of the considered 38 molecules are given in the above table-1.
2.2. Determination of Characteristic Polynomial:
The polynomials containing x as variable are derived from the adjacency
matrices of the molecules by the determinant evaluation method. All the polynomials for
the molecules are given in table-1.
Previously some scientists111-114 had derived recurrence relation for linear polyenens LN
with N vertices whose characteristic polynomials are symbolized by P(LN,x). The
polynomials up to L20 are given below:
P (L0; x)=1
P (L1; x)=x
P (L2; x )=x2-1
P (L3; x)=x3-2x
P (L4; x)=x4-3x2+1
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
32
P (L5; x)= x5-4x3+3x
P (L6; x)=x6 –5x4 +6x2-1
P (L7; x)= x7-6x5+10x3-4x
P (L8; x)= x8-7x6 +15x4-10x2+1
P (L9; x)= x9-8x7+21x5-20x3+5x
P (L10; x)= x10-9x8+28x6 -35x4+15x2-1
P (L11; x)= x11-10x9+36x7-56x5+35x3-6x
P (L12; x)= x12-11x10+45x8-84x6 +70x4-21x2+1
P (L13; x)= x13-12x11+55x9-120x7+126x5-56x3+7x
P (L14; x)= x14-13x12+66x10-165x8+210x6 -126x4+28x2-1
P (L15; x)= x15-14x13+78x11-220x9+330x7-252x5+84x3-8x
P (L16; x)= x16-15x14+91x12-286x10+495x8-462x6 +210x4-36x2+1
P (L17; x)= x17-16x15+105x13-364x11+715x9-792x7+462x5-120x3+9x
P (L18; x)= x18-17x16+120x14-445x12+1001x10-1287x8+924x6 -330x4+45x2-1
P (L19; x)= x19-18x17+136x15-560x13+1365x11-2002x9+1716x7-792x5+165x3-10x
P (L20; x)= x20-19x18+153x16-680x14+1820x12-3003x10+3003x8-1716x6 +495x4-55x2+1
A recurrence relation for computing P (LN; x) has also been suggested:
P (LN; x)= x P (LN-1; x)- P(LN-2; x)
(31)
For example:
The polynomial for linear polyene L5 can be obtained by equation (31).
P (L5; x)= xP (L4; x)- P (L3; x)
(32)
Where, P (L4; x)=x4-3x2+1
P (L3; x)=x3-2x
Putting these values in equation (32)
P (L5; x)= x(x4-3x2+1)-(x3-2x)
= x5-4x3+3x
Cyclic CN may be used to represent the carbon skeletons for cycloalkanes and [N]
annulenes. To compute P (CN; x) a recurrence relation has been obtained
P (CN; x)= P (LN; x)-P (LN-2; x)-2
(33)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
33
Thus the polynomial for the cyclobutane, C4, can be evaluated by equation (33).
P (C4; x)=P (L4; x)-P (L2; x)-2
(34)
Where, P (L4; x)=x4-3x2+1,
P (L2; x)=x2-1
Putting the above values in the equation (34)
P (C4; x)= (x4-3x2+1) – (x2-1) –2
=x4-4x2
A recurrence formula for computing the polynomial of the side chain-extended
graph of a cyclic system has been derived. The side chain extension is mostly due to a
single bond extension at any vertex of the graph. An isolated single bond is due to a
connection of two atoms (vertices). The polynomial of a side chain-extended graph of a
cyclic system (GN) can be represented as P (GN; x). The recurrence relation is
P(GN; x)=
P(G N ; x )( x 2 − 1)
-axr +dGN
x
(35)
Where, “a” is the degree of the carbon where the new bond is added when the number of
carbon atoms is >4 otherwise zero.
r is the power of last even term when the number of carbon atoms is even and last odd
term when the number of carbon atoms is odd.
dGN is the determinant of the GN graph.
In the equation (35) the constants and the negative power of x of the first term are
neglected. In the chain-extension process initially a single carbon atom (x) is added to the
ring by a naked bond. A naked bond can be represented as P (G2; x)/x2, where x
represents individual atom. For the addition of an atom (x) to a cyclic graph P (GN; x) the
addition should be
P (GN; x) × P (G2; x)/x2 ×x
Where, P (GN; x) represents the cycle, x represents the atom and P (G2; x)/x2 represents
the naked bond.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
34
For example:
The polynomial of
P (G5;x)=
can be derived from
by equation (34).
P(G 4 ; x ) × ( x 2 − 1)
-2x+dGN
x
(36)
Where the constant term and the terms having negative power of x are neglected from the
first term of the equation (36).
P (G5; x)=
( x 4 − 5x 2 + 4 x ) × ( x 2 − 1)
-2x+dG5
x
=(x5-6x3-4x2+5x-4)-2x+dG5
(37)
Neglecting the constant terms from (37)
P (G5; x) = (x5-6x3+4x2+5x)-2x+dG5
=x5-6x3+4x2+3x+dG5
0 1 0 0 0
1 0 1 0 1
Again, dG5= 0 1 0 1 1 =(-2)
0 0 1 0 1
0 1 1 1 0
∴P (G5; x)=x5-6x3+4x2+3x-2
The antithetic analysis of a cyclic graph reveals that a cycle can be generated from
prestruct with a single bond disconnection i.e. from a single species or may be with two
bond disconnections i.e. from two isolated species. An addition of a naked bond to two
atoms for the former case can lead to a cyclic product. For the later class of prestruct, the
isolated species may be an atom (vertex) or may be a C2 or more ordered graph with two
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
35
terminal atoms (vertices) to be bonded. In either case two bonds are to be connected with
no addition of atoms.
To determine the polynomial a recurrence formula can be proposed.
P (GN; x)=P (GN-2; x) P (G2; x)-2xN-2+axr+dGN
(38)
Where P (GN-2; x) itself represents a complete graph
P (G2; x) is the polynomial of a C2 graph.
a is zero for number of carbon atoms ≤4 or is equal to 3 and 1 alternatively when
number of carbon atoms > 4.
r is the power of the last even term when number of carbon atoms is even and of the last
odd term when the number of carbon atoms is odd.
dGN is determinant of the GN graph.
In the above equation the constant term of the first term is neglected.
For example:
P (G3; x)=P (G1; x)×P (G2; x)-2x+0+dG3
= x (x2-1)-2x+2= x3-3x+2
P (G4; x)=P (G2; x)×P (G2; x)-2x2+0+dG4
= (x2-1) (x2-1)-2x2 = x4-4x2+1
Neglecting the constant term from the first term P (G4; x)=x4-4x2
P (G5; x)=P (G3; x) P (G2; x)-2x3+3x+dG5
=(x3-2x) (x2-1)-2x3+3x+2=x5-5x3+5x+2
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
36
P (G5; x)=P (G3; x)P (G2; x)-2x3+x+dG5
=(x3-3x+2) (x2-1)-2x3+x+0
Neglecting the constant term in the first term of the above equation
P (G5; x)=x5-6x3+2x2+4x
P (G6; x)=P (G4;x) P (G2; x)-2x4+3x2+dG6
=(x4-4x2) (x2-1) –2x4+3x2-1
=x6-7x4+7x2-1
The above procedure of deriving polynomial may help in synthesizing molecules in
silico. Patra et al115 have reported the building up of a reaction scheme through adjacency
matrix to explain the generation of some interstellar molecules with various reference
frame for the reactants and reagents. They have proposed probable mechanisms during
the generation of some interstellar molecules.
2.3. Determination of the Eigenvalues:
The roots or eigenvalues of variable x are calculated. All the roots of the
considered 38 molecules are given in the table-1. The sum over of the roots of each graph
is found to be zero indicating the validity of the method. For the bipartite or alternant
graph we get a pair of roots where as for nonbipartite or nonalternant graph we do not get
pairing in the eigenvalues.
For example;
For
*
And for
*
*
roots are x=0, ±1.41421
*
roots are x=-2, 1, 1.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
37
From the above-calculated roots of variable x, the eigenvalues with most negative
value is regarded as ground state orbital and the wave function is designated as Ψ1. Then
Ψ2,Ψ3are notified in increasing order. Each root of a molecule y number of coefficients
are calculated where y is same as the number of carbon atoms in a molecule. To calculate
the coefficient values first the cofactor polynomials are derived from its adjacency matrix
designated as C1, C2, CN. All the cofactor polynomials are then divided by the C1
polynomial and there after putting the corresponding x value we get C1/C1, C2/ C2, -----,
CN/C1. Then from these value we get (C1/C1)2, (C2/C1)2…………(CN/C1)2,∑(Ci/C1)2 and ,
(∑(Ci/C1)2)1/2 .Then the above calculated values of C1/C1,C2/ C2,-----,CN/C1 are divided
by (∑(Ci/C1)2)1/2 to get the values of coefficients C1’,C2’------------CN’. From the values
of coefficients and roots various parameters such as bond order, electron density, free
valence index, energy, graph angle etc. are calculated.
2.4. Determination of Bond Order:
The bond order of a molecule is calculated from its coefficient values by the formula.
Pab=n1(Ca)ψ1(Cb)ψ1+n2(Ca)ψ2(Cb)ψ2+-------------+nn(Ca)ψn(Cb)ψn
(39)
When n1, n2, --------, nn are number of electrons in 1st, 2nd, -----nth shell respectively. The
values of bond orders calculated from the above equation for all the considered molecules
are given in the table-2.
Table-2: Graph theoretical bond order values of some small organic molecules.
Graph
No.
Molecular structure
2
1
Number
Bond Orders
of pi
electrons
2
P12=1.0003
1
2
2
1
3
2
P12=0.7069, P23=0.7073
2
P12=P23=P13=0.666551
1
3
3
2
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
4
2
P12=P23=P24=0.577349
4
P12=0.886774,P23=0.453931,
P34=0.883404
4
P12=P23=P34=P14=0.5
4
P12=0.758295,P23=P24=0.458673,
P34=0.817554
4
P12=P23=P34=P14=P13=0.498319
4
P12=P23=P34=P14=P13=P24=1
4
P12=P23=P24=P25=0.5
4
P12=P45=0.788675, P23=P34=0.577345
4
P12=0.560457, P23=0.651527,
P35=P34=0.328016
4
P12=P23=0.610131, P34=P14=0.357408,
P45=0.862856
4
3
2
5
2
4
1
3
2
3
1
4
6
1
7
2
8
4
3
1
4
2
3
1
4
2
3
9
1
2
10
3
5
4
2
4
11
1
5
3
4
12
2
1
3
5
13
1
5
4
2
3
1
14
2
5
4
P12=P15=0.644891, P23=P45=0.199959,
P34=0.928354
4
3
1
15
2
5
3
4
4
P12=P15=0.701555, P23=P45=0.551942,
P34=0.977456, P25=0.538822
38
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
39
1
16
5
2
4
P12=P15=0.506149,
P23=P24=P35=P45=0.345784, P34=0.781979
3
4
1
17
2
5
P12=P15=0.313076, P34=0.384965,
P23=P45=P24=P35=0.447831
2
P12=0.439004, P23=0.431429,
P34=0.479917, P45=0.252726,
P24=0.610133, P14=0.479914
4
P12=0.793929, P23=P25=0.343464,
P34=P45=0.58821, P35=0.631675
2
P12=0.472345, P23=P24=P25=0.140917,
P34=P45=P35=0.129742
3
4
5
4
1
18
2
3
2
3
19
5
2
1
4
20
3
5
2
1
5
21
2
3
1
4
4
P12=P34=P23=P14=0.346219,
P13=0.782069, P25=P45=0.505228
6
P12=P23=P34=P45=P14=P25=-0.0000004,
P24=0.499997
4
P12=P34=0.776694, P23=0.165328,
P45=P15=0.3919299, P35=P25=0.525445
4
P12=0.816387, P23=0.41352,
P34=P35=0.600266, P45=0.694325
1
22
5
4
3
2
5
23
4
1
2
3
1
24
2
3
4
5
3
25
4
2
1
4
P12=P45=0.724572, P23=0.554702,
P34=0.556998, P24=0.362278
4
P12=P45=0.81063,
P13=P23=P34=P35=0.485071
5
2
26
1
3
4
5
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
2
27
40
3
2
P12=0.174182, P23=0.57531,
P34=P35=P36=0.441607
4
6
5
1
4
28
3
2
4
P12=P26=0.666666, P23=0.333333,
P34=P35=0
4
P12=0.760822, P23=P45=P46=0.615539,
P34=0.470218
6
P12=0.909231, P45=0.908147,
P23=0.407488, P34=0.408376,
P36=0.816244
6
P12=P56=0.87113, P23=P45=0.483432,
P34=0.784851
6
P12=0.797554, P34=P45=P56=0.550225,
P34=P36=0.446563
6
P12=P45=0.499977, P23=P56=P34=0.7071117,
P16=0.707076, P36=0.000007
5
6
5
29
2
4
6
3
1
4
2
3
30
5
1
6
3
31
6
4
2
1
32
33
5
5
4
6
3
1
2
5
4
6
3
1
2
1
34
2
6
3
5
4
6
P12=0.021745, P23=P26=0.651483,
P34=P56=0.275564, P45=1.224986
1
35
6
2
5
3
P12=P23=P34=P45=P56=P16=0.333333
6
4
2
1
36
6
5
6
3
4
P12=0.462983, P23=P34=0.611843,
P45=0.463026, P56=0.682228,
P36=0.102032, P24=0.453053,
P61=0.682369
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
2
1
37
3
6
6
P12=0.666664, P23=P24=0.33333,
P34=P56=0.833341, P45=P36=0.166669,
P16=P15=0.33555
6
P12=P13=0.142229, P23=1.14434,
P34=P25=0.144335, P45=0.644333,
P45=P56=0.642226
4
5
1
38
41
2
3
4
6
5
For convenience, during the calculation of bond order the electrons for double
bonds are only considered. The numbers of electrons considered in each case are given in
table 2. Hence the bond orders are due to pi-electron system only and are found to be 1
unit less than the value obtained by normal quantum chemical method. For example, the
bond orders for butadiene116 are reported to be 1.894 and 1.447, whereas in the present
case the values are found to be 0.887 and 0.454. Similarly for 3-methylene penta-1, 4diene, the bond orders obtained from quantum chemical calculation are found to be
1.930, 1.859 and 1.363 whereas the corresponding values obtained from graph theoretical
technique are0.909, 0.816 and 0.408 respectively.117 The values obtained in the present
case are referred to as graph theoretical bond orders (GTBO).
The trend in GTBO values depicts the resonance characteristics in a molecule.
Molecules with identical resonating structures have same GTBO for similar bonds. When
the delocalization is restricted there is a change in the GTBO values for the bonds. As in
case of 7, the GTBO values of 1,2 and 1,3 bonds are found to be 0.758 and 0.818
respectively, while the other two bonds have the value 0.459. Similar is the case of
for 26.
2.5. Determination of Electron Density:
The closed shell electron density of a molecule is calculated from its coefficient values by
the formula
qa=n1 (Ca)2+n2 (Ca)2+n3 (Ca)2+ ----------- +nn (Ca)2
(40)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
42
When n1, n2, ----nn are number of electrons in 1st, 2nd, ------nth shell respectively. The
values of electron densities and there summation for all considered molecules are given
in table-3.
Table-3: Electron densities of some organic molecules calculated using closed shell
model
N
Graph
q1
q4
q2
q3
q5
q6
qi
∑
No.
q =1
1
1.0003
1.0003
2.0006
2
1.000264
0
1.000264
2.000528
3
0.666551
0.666551
0.666551
1.999653
4
0.333331
1
0.333335
0.333335
2.000001
5
0.99495
1.083691
1.083691
0.998029
4.160361
6
1.5
0.5
1.5
0.5
4
7
1.480864
0.876748
0.817554
0.817554
3.99272
8
1.541267
0.459035
1.541267
0.459035
4.000604
9
1
1
1
1
4
10
0.25
1
0.25
0.25
0.25
2
11
0.666667
1
0.66667
1
0.66667
4.000004
12
1.20548
1.041097
0.767129
0.493145
0.49314
3.999996
13
0.500001
1
0.500001
1
1
4.000002
14
1.191814
0.47574
0.928354
0.928354
0.47574
4.000002
15
0.967445
0.538822
0.977456
0.977456
0.538822
3.994209
16
1.535518
0.450264
0.781979
0.781979
0.450264
4.000004
17
0.188145
0.520963
0.384965
0.384965
0.520963
1.999971
18
0.339348
0.548488
0.339353
0.678706
0.094106
2.000001
19
0.832192
0.956843
0.631675
0.749062
0.63167
3.801447
20
1.45772
0.153054
0.129742
0.129742
0.12974
2
21
0.782074
0.449959
0.78207
0.449959
1.53593
3.999992
22
2.93329
0.6
0.9333384
0.6
0.933338
5.9999668
23
0.969359
0.718119
0.718121
0.969364
0.62504
3.999999
24
0.847846
1.034595
0.728915
0.694325
0.69432
4.000006
25
0.844047
0.915072
0.481763
0.915068
0.84404
3.996418
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
43
26
0.81063
0.81063
0.757466
0.81063
0.81063
3.999986
27
0.08397
0.36132
0.91603
0.212894
0.21289
0.212894
2.000002
28
0.5
0.999999
0.999999
0.5
0.5
0.5
3.999998
29
0.599942
1.000029
0.600018
1
0.40001
0.400006
4.000001
30
0.999748
1.001062
0.999939
0.99997
0.99978
0.999506
6.000004
31
1.000008
1.000008
1
1
1.00001
1.000008
6.000032
32
1.333334
1.000007
1.333332
0.499996
1.33333
0.499996
5.999998
33
0.999983
0.999995
1.00004
1.000012
1.00004
0.999946
6.000019
34
1.136745
1.2075
0.66288
1.224986
1.22498
0.66288
6.119977
35
0.999999
0.999999
0.999999
0.999999
0.99999
0.999999
5.999994
36
1.147195
1.005789
0.556915
1.005899
1.14705
1.13716
6.000006
37
1.333332
1.333332
0.833338
0.833338
0.83334
0.833338
6.000016
38
1.211324
1.14434
1.14434
0.644333
0.64433
1.211324
5.999994
The open shell electron densities of a molecule are calculated by considering at
least one electron at each valence state.
Table 4: Electron densities of some small organic molecules using open shell model
Graph
No.
1
q1
q2
q3
q4
q5
q6
N
∑ qi
q =1
1.000264
1.000264
2.000528
2
0.999732
1.000264
1.000332
3.000328
3
1.666587
0.666607
0.666607
2.999801
4
1.666666
5
0.99459
6
1.5
7
2.744032
8
2.081267
9
1
1
10
1.750002
1
0.416666
0.416666
0.416666
4
11
0.999999
1
0.999999
1
0.999999
5
12
1.205482
0.767128
0.493145
0.493145
1
0.666669
0.66669
4.000004
1.003691
0.998029
4.000001
1.5
0.5
4
0.438414
0.408777
0.408777
4
2.081267
0.418885
0.418885
1.003691
0.5
1.041097
1
5.000304
1
4
3.999997
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
13
1
1
1
0.999999
44
1
5
5
14
1.738614
0.82441
0.80628
0.80628
0.82441
15
2.8002
0.551306
0.548588
0.548588
0.551306
16
2.418177
0.581064
0.709844
0.709844
0.581064
5
17
2.094072
0.260482
0.192482
0.192482
0.260482
3
18
0.919664
0.420695
0.919692
0.839347
0.900612
4.00001
0.5
4
19
20
21
1
1.377309
1
0.5
0.35937
0.087775
0.499996
1
22
1
1
1
4.999988
0.087775
0.087775
2.000004
0.4999996
0.999996
3.999988
0.499997
0.500002
0.499997
0.500002
2.999998
23
0.861804
0.638188
0.638188
0.861804
0.999998
3.999982
24
0.999996
1.00001
0.999994
0.5
0.5
4
25
0.999993
0.99999
0.999992
0.99999
1.004501
5.004466
26
0.499992
0.500004
1.000008
0.500002
0.500002
3.000008
27
0.999993
0.999995
1.000006
0.333336
0.333336 0.333336 4.000074
28
1.5
0.999999
0.999999
1.5
29
1.399943
1.000029
1.400019
0.999999
0.600006 0.600006
30
0.999745
1.001063
0.999933
0.99997
0.999748 0.999506 5.999968
31
0.993275
1.103704
0.965897
0.965897
1.103704 0.993275 6.125752
32
1.333335
1.000007
1.333333
0.499996
1.333333 0.499996
33
0.999983
0.999995
1.00004
1.000012
1.000043 0.999946 6.000019
34
1.623182
1.227593
0.830073
0.744533
0.744533 0.830073 6.048324
35
0.999999
0.999999
0.999999
0.999999
0.999999 0.999999 5.999992
36
1.000036
0.999975
0.999999
1.000032
0.999962 0.999995 5.999999
37
0.999999
0.999999
0.500005
0.500005
0.500005 0.500005 4.000018
38
1.249997
0.749999
0.749999
0.749999
0.749999 1.249997 5.49999
1.5
1.5
8
6
6
The charge densities on each atom were calculated by using quantum mechanical
models on graph theoretical results. The values are found to be in the same trend as in
case of bond order. However, the antiaromatic characteristics of the cyclobutadiene can
be significantly marked from the charge density values. The values at alternate carbons
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
45
are found to be same and different for adjacent atoms. For 4 and 10 the central carbon has
the highest values (=1) and the other carbon atoms share equal values sum totaling to 1.
In the open shell method, an equal distribution of electrons is assumed for each carbon
atom and thus a change in the vales are obtained in a few cases. A significant change has
been observed in 4, where the central carbon retains its value (=1) while one of the atoms
has a value one unit more than other two carbon atoms.
2.6.Free Valence Index:
The free valence index of a carbon atom of a molecule is calculated by the formula
σ
σ
σ
Π
Π
Fa= 4.732-[m× PCa
− H + r × PCa − Cb + s × PCa − Cc + q × PCa − Cb + o × PCa − Cc ]
(41)
Where, m=number of Ca-H bond
r=number of Ca-Cb σ-bond
s=number of Ca-Cc σ-bond
q=number of Ca-Cb π-bond
o=number of Ca-Cc π-bond
σ
σ
σ
Π
Π
and PCa
− H , PCa − Cb , PCa − Cc , PCa − Cb , PCa − Cc represent the bond order of corresponding bonds.
The values are given in table-5.
Table 5: Free valence index at different atoms of some small organic molecules.
Molecular structures
2
1
F1
0.729
F2
0.729
F3
1.0251
0.3178
1.0297
-0.6011
3.0651
0.398898
0.8452
0.3913
0.3947
F4
2
1
3
1
2
3
2
1
4
3
0.8486
F5
F6
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
46
1
2
0.1547
-0.00005
1.1547
0.1547
0.732
0.732
0.732
0.732
0.9737
0.0564
0.4561
0.4561
0.7354
0.2763
0.7353
0.2763
0.232
0.232
0.232
0.232
1.232
-1.268
0.232
0.232
0.232
0.9433
0.366
0.5773
0.366
-0.0567
0.7645
0.5117
0.7645
0.1543
-0.1309
-0.5578
0.8872
0.6037
0.6037
0.8872
-0.6711
-0.0603
0.2026
0.2026
-0.0603
-0.2803
0.5343
0.2585
0.2585
0.5343
0.1059
-0.7898
0.4514
0.4514
-0.7898
4
3
2
3
1
4
1
2
4
3
1
4
2
3
1
4
2
3
1
2
5
3
4
2
4
1
5
3
1
5
4
2
3
1
2
5
3
4
1
5
2
3
4
1
5
2
4
3
1
5
4
2
3
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
5
4
1
47
0.8131
0.2514
-0.1794
-1.0907
0.4793
0.9381
0.2511
0.1687
0.5556
0.1687
0.2597
-0.1631
0.3316
1.3316
1.3316
1.1715
0.5200
0.4244
0.404
1.404
0.2575
0.5343
0.2575
0.5343
-0.2785
-0.068
-1.068
-0.068
-1.068
-0.68
0.5634
0.2645
0.2645
0.5634
-0.1028
0.9156
0.5021
0.118
0.4374
-0.5626
1.0074
0.0905
-0.3797
0.0882
1.0074
0.4363
0.4363
0.4363
0.4363
-1.2083
0.5772
1.0829
0.5461
0.6314
0.6314
3
2
3
5
2
1
4
3
5
2
1
4
2
1
3
5
5
2
3
1
4
1
5
4
3
2
5
4
1
2
3
1
2
3
4
5
3
4
2
1
5
1
2
3
4
5
2
1
3
6
4
5
0.6314
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
48
4
3
2
1.0653
0.0653
1.3987
1.732
0.732
0.0653
0.9712
0.3556
0.6462
0.6462
0.1165
0.1165
0.8228
0.4153
0.0999
0.4155
0.8239
0.9158
0.8609
0.3774
0.4637
0.4637
0.3775
0.8608
0.9345
0.3842
0.2886
0.7352
0.6316
0.7352
0.525
0.525
0.3178
0.525
0.5245
0.3178
1.1711
0.4073
0.805
0.2315
0.2315
0.805
-0.268
-0.268
-0.268
-0.268
-0.268
-0.268
0.5867
0.2041
0.4063
0.2041
0.5867
0.2653
0.3942
0.3987
0.3987
0.3987
0.3964
0.3964
5
6
5
2
4
6
3
1
4
2
3
5
1
6
3
6
4
2
1
5
4
6
3
1
2
5
4
6
3
1
2
5
1
2
6
3
5
4
1
6
2
5
3
4
2
1
6
3
4
5
2
1
6
5
3
4
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
1
49
2
3
1.4475
0.3011
0.3011
0.3011
0.3011
0.4476
4
6
5
2.7. Graph Energy:
The energy of a molecule is equal to the sum over of absolute values of its
eigenvalues.
n
E = ∑ λi .
(42)
i =1
The calculated energy for each of the 38 molecules is given in the table-6
2.8. Graph theoretical descriptors m, K and α
Some molecular descriptors m, K and α are calculated from eigenvalues by the
formulas:
n
2
∑ λ = 2m
(43)
i =1
K=
2m + n − 2E
2m + n + 2E
α=
2K 2
1− K2
as, 2E ≤ (2m+n)
[ where, n=number of carbon atoms]
(44)
(45)
The values of these above parameters for a set of 38 molecules are given in the table-6.
2.9.Angle of graph energy
From the values of E and m then the angle of graph energy i.e. θ is calculated by
the formula:
cos θ =
E
2mn
(46)
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
50
The values of θ for a set of 38 molecules are given in the table-6.
Table-6: Graph energy, graph angle, M, K and α of some organic molecules
Structure
E
2
2
1
θ0
M
1
0
α
K
0
0
2
1
3
2.8
2
35
0.106121
0.02278
4
3
19.5
0.058824
0.006944
4.5
3
23.3
0.052632
0.005555
3.5
3
44.4
0.176471
0.064286
4
4
45
0.2
0.083
1
2
3
2
4
1
3
1
2
4
3
2
3
1
4
1
2
4
3
1
4
2
3
1
4
2
3
4.5
3.4
30.8
0.093238
0.01754
5.1
5
35.9
0.15482
0.04916
6
6
30
0.142857
0.041666
4
4
50.8
0.238095
0.120193
5.5
4
29.6
0.083333
0.013983
5.6
5
37.6
0.145038
0.042956
6.5
5
23.2
0.071429
0.010256
1
2
3
5
4
2
4
1
5
3
1
5
4
2
3
1
2
5
3
4
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
51
1
5
2
3
6.3
6
35.6
0.148649
0.502742
6
6.2
40.4
0.183674
0.069828
6.7
7.1
37.33
0.177914
0.065376
6
6
39.2
0.172414
0.061275
6.7
6
30
0.118421
0.028446
7
7
33
0.151515
0.046993
5.2
4
34
0.108624
0.02388
6.4
7
40
0.194969
0.07903
6
7
44
0.225806
0.107456
7
7
33
0.151515
0.046993
6.4
5
24.6
0.076919
0.011904
4
1
5
2
3
4
1
2
5
3
4
5
4
1
3
2
3
4
5
2
1
4
3
5
2
1
4
2
1
3
5
5
2
3
1
4
1
5
4
3
2
5
4
1
2
3
1
2
3
4
5
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
52
3
4
2
1
5.8
5
35
0.168142
0.058189
7
6
25.4
0.096774
0.018907
5.8
5
41.6
0.15942
0.52155
6
5
39
0.142857
0.041666
7.3
6.7
35.5
0.141177
0.040673
7
5
25.4
0.066667
0.008929
7
5
25.4
0.066667
0.008929
6.6
6
39
0.153846
0.048485
7.7
7
33
0.129944
0.034351
7.5
6
33
0.090909
0.016667
8
6
19.5
0.058824
0.006945
5
1
2
3
4
5
1
2
3
4
6
5
1
4
3
2
5
6
5
2
4
6
3
1
4
2
3
5
1
6
3
6
4
2
1
5
4
6
3
1
2
5
4
6
3
1
2
5
1
2
6
3
5
4
1
6
2
5
3
4
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
53
2
1
3
6
5
4
1
2
3
6
8.2
8
33
8
9
40
8.3
8
32
0.145833
0.2
0.043459
0.083333
4
5
1
2
3
0.139896
0.039923
4
6
5
E values are found to be associative on the graphical fragments of the structure.
Thus for each fragment a value can be assigned for the operation engendering the
corresponding E value. In general, for acyclic system a value of 2 can be assigned to an
isolated olefinic double bond and 0.8 to an isolated single bond. For cyclic systems it also
holds good for all the concerned cycles other than four membered cyclic derivatives. The
typical four membered compound, cyclobutadiene, is an antiaromatic compound where
4n cyclic pi-electron system destabilizes the molecule. Thus in 4 membered cyclic
derivatives a substantial decrease in the determined E value is found from the calculated
values using fragmentation method. For example, graph “4” correspond to
cyclobutadiene, where the E value can be calculated to be 5.6 which is determined to be
4. The decrease of 1.6 values may be attributed to the antiaromaticity. It can be
generalized for the other cyclobutadiene derivative.
The parameters M, θ, K and α are reported to be molecular descriptors and have
applicability in quantitative structure property/activity relationships. At present, due to
unavailability of data on the molecular characteristics no attempts have been made for
using these parameters.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
54
3. REFERENCES
1.
N. J. Turro, Angew, Chem. Int. Ed., Engl, 25 (1986), 882.
2.
N. Trinajstic, in MATH/CHEM/COMP 1987, Ed. R. C. Lacher, Elsevier,
Amsterdam, 83 (1988).
3.
M. Randić, Ind. J. Chem., 42A (2003), 1207.
4.
R.J. Wilson, Introduction of Graph Theory, Oliver and Boyd Edinburg,
(1972).
5.
D.E. Johnson, and J.R. Jhonson, Graph Theory with Engineering Application,
Ronald Press, New York, (1972).
6.
J.W. Essam, and M.E. Fisher, Rev. Mod. Phy. 42(1970), 272.
7.
F. Harary, Graph Theory, Addison-Wesley, Reading MA, (1971) 2nd printing.
8.
J. Lederberg, Proc. Natl. Acad. Sci, U.S.A., 53 (1965) 134.
9.
V. Prelog, Nobel Lecture, Stockholm, (1975).
10.
V. Prelog, Science, 17 (1976) 193.
11.
A.T. Balaban, J. Chem. Inf. Comput. Sci,. 25 (1985) 334.
12.
L. Pogliani, Ind. J. Chem., 42A (2003) 1347.
13.
E.V. Konstantinova, V.A. Skorobogatov and M.V. Vidyuk, Ind. J. Chem., 42A
(2003) 1227.
14.
A. Perdih and B.Perdih, Ind. J. Chem., 42A (2003) 1219.
15.
A. Perdih, Ind. J. Chem., 42A (2003) 1246.
16.
I. Gutman, D. Vidović and B. Furtula, Ind. J. Chem., 42A (2003) 1241.
17.
L. Türker, Ind. J. Chem., 42A (2003) 1295.
18.
F. Torrens, Ind. J. Chem., 42A (2003) 1258.
19.
O. Ivanciuc, and D.J. Klein, J. Chem. Inf. Comput. Sci., 42 (2002) 8.
20.
D.J. Klein, Ind. J. Chem., 42A (2003) 1264.
21.
A.A. Dobrynin, Ind. J. Chem., 42A (2003) 1270.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
55
22.
I. Gutman, D. Vidovic, B. Furtula, Ind. J. Chem., 42A (2003) 1272.
23.
B. Lucic, A. Milicevic, S. Nikolic, N. trinajstic, Ind. J. Chem., 42A (2003)
1279.
24.
H.C. Longet-Higgins, Mol. Phy. 6 (1963) 445.
25.
L. Bytatus, D.J.Klein, International Journal Of Quantum Chemistry, 70 (1998)
215.
26.
M.O. Ivancius, L. Bytatus, D.J. Klein, J. Chem. Phys., 116 (2002) 4735.
27.
L. Bytatus, D.J. Klein, J. Phys. Chem. A, 103 (1999) 5196.
28.
E. Estrade, G. Patlewicz and E. Uriarte, Ind. J. Chem., 42A (2003) 1315.
29.
R. Natarajan, P. Kamalakanan and I. Nirdosh, Ind. J. Chem., 42A (2003) 1330.
30.
P. Duchowicz, R.G. Sinani, E.A. Castro and A.A. Toropov, Ind. J. Chem., 42A
(2003) 1354.
31.
V.K. Sharma, Romi and S. Kumar, Ind. J. Chem., 42A (2003) 1379.
32.
S.S. Basak, D. Mills, M. Mumtaz and K. Balasubramanian, Ind. J. Chem., 42A
(2003) 1385.
33.
S. Gupta, M. Singh and A.K. Madan, Ind. J. Chem., 42A (2003) 1414.
34.
V.K. Agrawal, S. Kamarkar, P.V. Khadikar, S. Shrivastava and I. Lukovits,
Ind. J. Chem., 42A (2003) 1426.
35.
P.V. Khadikar, I. Lukovits, V.K. Agrawal, S. Shrivastava, M. Jaiswal, I.
Gutman. S. Karmarkar and A. Shrivastava, Ind. J. Chem., 42A (2003) 4136.
36.
S. Singh, B.K. Mishra and R.K. Mishra, Ind. J. Chem., 27A (1988) 188.
37.
R.K. Mishra, S. Singh and B.K. Mishra, Ind. J. Chem., 28A (1989) 927.
38.
M. Kuanar, R.K. Mishra and B.K. Mishra, Ind. J. Chem., 35A (1996) 1026.
39.
M. Kuanar and B.K. Mishra, J. Serb. Chem. Soc., 62 (1997) 289.
40.
M. Kuanar, B.K. Mishra and I. Gutman, J. Serb. Chem. Soc., 62 (1997) 1025.
41.
M. Kuanar, S. Kuanar and B.K. Mishra, Ind. J. Chem., 38A (1999) 525.
42.
M. Kuanar, B.K. Mishra and I. Gutman, Ind. J. Chem., 40A (2001) 4.
43.
Y. Pradhan, M. Kuanar and B.K. Mishra, Ind. J. Pharm. Sc., 62 (2000) 133.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
56
44.
M. Kuanar, S. Kuanar, S. Patel and B.K. Mishra, Ind. J. Pharm. Sc., 42A
(2003) 2463.
45.
M. Kuanar and B.K. Mishra, Ind. J. Chem., 36A (1997) 571.
46.
M. Kuanar, S. Kuanar and B.K. Mishra, Ind. J. Chem., 38A (1999) 113.
47.
M. Kuanar and B.K. Mishra, Bull. Chem. Soc. Jap. 71 (1998) 291.
48.
M. Randic, SIAM J. Aig. Disc. Math. 69(1985)145.
49.
K. Balasubramaniam. in Mathematics and Computational Concepts in
Chemistry, N. TrinajsticEd., Horwood, Chichester(1980)20.
50.
N. Trinajstic, D. J. Klein and M. Randic, Int. J. Quantum Chem.: Quantum
Chem.Symp., 20 (1986) 699.
51.
D. Berman and K. W. Holladay, J. Math. Chem., 1 (1987) 405.
52.
R.B. King, J. Math. Chem.., 2 (1988) 89.
53.
I. Gutman, And O.E. Polansky, Mathematical Concepts in Organic Chemistry,
Sprin-ger-verlag, Berlin (1986).
54.
O.E. Polansky, in Chemical Graph Theory: Introduction and Fundamentals,
D. Bonchev, and D.H. Rouvray, Eds. Abacus Prees/Gordon and Breach, New
York, 41(1991).
55.
. R.B. Mallion, R.B.,Schwenk, A.J., and N. Trinajstić, Croat.. Chem.Acta., 46
(1974) 71
56.
R.B. Mallion, N. Trinajstić, and A.J. Schwenk, Z. Naturforsch. 29a (1974)
1481.
57.
R.B. Mallion, N. Trinajstić, and A.J. Schwenk, in recent Advances in Graph
Theory, M. Fiedler, Ed., Academia, Prague, (1975) 345.
58.
A. Graovac, O.E. Polansky, N.Trinajstić, N. Tyutyulkov, and Z. Natuforsch.,
1975,30a,1696.
59.
M.J. Righy, R.B. Mallion, and A.C. Day, Chem. Phys. Lett. 51 (1977) 178;
erratum Chem. Phys.Lett. 53 (1978) 418.
60.
R.B. Mallion, in Applications of combinatorics, R.J.Wilson, Ed., Shiva
Publishing, Nantwich, Cheshire, U.K., (1982) 87.
61.
H.H. Schmidtke, J. Chem. Phys. 45 (1966) 3920.
62.
H.H. Schmidtke, Coord. Chem. Rev., 2 (1967) 3.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
57
63.
A. Graovac, and N. Trinajstić, Math. Chem. (Mülheium/Ruhr), 1 (1975)
159,erratum Math. Chem. (Mülheim/Ruhr), 5 (1979) 290.
64.
E. Heilbronner, Tetrahedron Lett. (1964) 1923.
65.
A. Graovac, and N. Trinajstić, Croat. Chem. Acta, 47 (1975) 95.
66.
A. Graovac, and N. Trinajstić, J. Mol. Struct. 30 (1976) 416.
67.
I. Gutman, Z. Naturforsch 33a (1978) 214.
68.
D.H. Rouvray, in Chemical Applications of Graph Theory, A. T. Balaban.,
Ed.,Academic Press, London(1976) 175.
69.
N. Trinajstic, in Semiempirical Methods of Electronic Structure Calculation.
Part.A. Techniques,. Vol. 7, Modern Theoretical Chemistry, A. G. Segal., Ed.
Plenum Press, New York (1977) 1.
70.
D.H. Rouvray, in Mathematics and Computational Concepts in Chemistry, N.
Trinajstić, Ed., Horwood, Chichester, (1986) 295.
71.
S.L. Hakimi, and S.S. Yau, Q. Appl.Math. 22 (1964) 305.
72.
M. Barysz, G. Jashari, R.S. Lall, V.K., Srivastava, and N. Trinajstić, in
Chemical Applications of Topology and Graph Theory, R.B King, Ed.,
Elsevier, Amstredam, (1983) 222.
73.
N., Trinajstić, Kem. Ind (Zagreb), 33 (1984) 311.
74.
M. Randić, A. Sabljić, S. Nikolić, and N. Trinajstić, Int. J. Quantum Chem.
Syamp. 15 (1988) 267.
75.
J.K. Senior, J. Chem. Phys., 19 (1951) 865.
76.
Z. Slanina, Contemporary Theory of Chemical Isomerism, Academia, Praha
(1986).
77.
G.C. Pimentel, R.C. Spartley, Understanding Chemistry, Holden-day, San
Francisco (1971) 733.
78.
A. Cayley, Ber. Dtsch. Chem. Ges. 8 (1857) 1056.
79.
C. Domb, in Phase Transitions and Critical Phenomena, 3, C. Domb, and
Green, M.S., Eds. Academic Prees, London (1974) 1.
80.
A. Cayley, Philos. Mag., 13 (1857) 172.
81.
G. Kirchhoff, Ann. Phys, Chem., 70 (1847) 497.
82.
C. Jordan, J. Reine Angew. Math. 70 (1869) 185.
83.
A.Cayley, J. Rep. Br. Assoc. Adv. Sci., 45 (1875) 257.
84.
A.Cayley, Am. J. Math. 4 (1881) 266.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
85.
A.Cayley, Philos, Mag., 47 (1874) 444.
86.
H. Schiff, Ber. Dtsch. Chem. Ges. 8 (1875) 1542.
87.
H.R. Henze, and C.M. Blair, J. Am. Chem. Soc., 53 (1931) 3042.
88.
H.R. Henze, and C.M. Blair, J. Am. Chem. Soc., 53 (1931) 3077.
89.
H.R. Henze, and C.M. Blair, J. Am. Chem. Soc., 54 (1932) 1098.
90.
H.R. Henze, and C.M. Blair, J. Am. Chem. Soc., 54 (1932) 1538.
91.
H.R. Henze, and C.M. Blair, J. Am. Chem. Soc., 55 (1933) 680.
92.
L. Bytautas and D. J. Klein, CCACAA, 73(2) (2000) 331.
93.
L. Bytautas and D. J. Klein, J. Chem. Inf. Comp. Sci. 38 (1998) 1063.
94.
L. Bytautas and D. J. Klein, J. Chem. Inf. Comp. Sci. 39 (1999) 803.
95.
L. Bytautas and D. J. Klein, J. Chem. Inf. Comp. Sci., 40 (2000) 471.
96.
L. Bytautas and D. J. Klein, Theor. Chem. Acc., 101 (1999) 371.
97.
L. Bytautas and D. J. Klein, Phys. Chem.-Chem. Phys., 1 (1999) 5565.
98.
L. Bytautas and D. J. Klein and T.G. Schmalz, New J. Chem. (2000)
99.
D. J. Klein, and O. Ivancius, J. Math. Chem., 3 (2001) 30.
58
100. A. Graovac, I. Gutman, N. Trinajstić, Topological approach to the Chemistry
of conjugated molecules (Springer, Berlin) (1977).
101. J.R. Dias, Molecular Orbital Calculations Using Chemical Graph Theory
(Springer, Berlin) (1993)
102. S. Markovic, Ind. J. Chem., 42(2003) 1304.
103. S.C. Rakshit, M. Banerjee and B. Hazra, Ind. J. Chem., 27A (1988), 183.
104. S.C. Rakshit and B. Hazra, J. Ind. Chem. Soc. 67 (1990), 887.
105. M. Randic, M.Vracko and M. Novic, in QSAR/QSPR studies by Molecular
Descriptor, edited by M. V. Diudea(2001) pp-147.
106. I. Gutman, Ber Math-Stat Sekt Forschungszentrum Graz, 103 (1987) 1.
107. L. Türker, J. Mol. Stru. 578 (2002) 191.
108.
L. Türker, J. Mol. Stru. 584 (2002) 183.
109. N.V. Efimov, E.R. Rozendron, Linear Algebra. And Multidimensional
Geometry, Mir, Moscow (1975).
110. I.V. Proskuryakov, Problems in Linear Algebra. Mir, Moscow (1978).
111. H. Hosoya, Theoret. Chem. Acta. 25 (1972) 215.
112. F. Harary, C. King, A. Mowshowitz and R. C. Read, Bull. London Math. Soc.,
3 (1971) 321.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
59
113. A. J. Schwenk, in Graphs and Combinatorics, R. Beri and F. Harary, Eds.,
Springer-Verlag, Berlin (1974) 153.
114. E. Heilbronner, Hliv. Chim. Acta, 36 (1953) 170.
115. S. Patra, R.K.Mishra, B.K.Mishra, Intern. J. Quantum Chem., 62 (1997) 495.
116. J. March, in Advanced Organic Chemistry: Reactions, Mechanism and
Structure, John Wiley Inter Science, New York (1999) pp. 33.
117. Ibid pp.34.
GRAPH CHARACTERISTICS OF SOME HYDROARBONS
60