Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2601-2627
© Research India Publications
http://www.ripublication.com/gjpam.htm
Some Computational Aspects for the
Line Graph of Banana Tree Graph
Muhammad Saeed Ahmad
Department of Mathematics,
Government Muhammdan Anglo Oriental College,
Lahore 54000, Pakistan.
Waqas Nazeer
Division of Science and Technology,
University of Education,
Lahore 54000, Pakistan.
Shin Min Kang1
Department of Mathematics and RINS,
Gyeongsang National University,
Jinju 52828, Korea.
Chahn Yong Jung
Department of Business Administration,
Gyeongsang National University,
Jinju 52828, Korea.
Abstract
A line graph has many useful applications in physical chemistry. M-polynomial
is rich in producing closed forms of many degree-based topological indices which
correlate chemical properties of the material under investigation. In this report,
we compute closed form of the M-polynomial for the line graph of banana tree
1
Corresponding author.
graph. From the M-polynomial we recover some degree-based topological indices.
Moreover, we plot our results.
AMS subject classification: 05C12, 05C90.
Keywords: M-polynomial, topological index, line graph.
1.
Introduction
In chemical graph theory, a molecular graph is a simple graph (having no loops and
multiple edges) in which atoms and chemical bonds between them are represented by
vertices and edges respectively. A graph G(V , E), with vertex set V (G) and edge set
E(G) is connected if there exists a connection between any pair of vertices in G. A
network is simply a connected graph having no multiple edges and loops. The degree
of a vertex is the number of vertices which are connected to that fixed vertex by the
edges. The distance between two vertices u and v is denoted as d(u, v) = dG (u, v) and
is defined as the length of shortest path between u and v in graph G. The number of
vertices of G, adjacent to a given vertex v, is the “degree” of this vertex, and will be
denoted by dv (G) or, if misunderstanding is not possible, simply by dv . The concept of
degree is somewhat closely related to the concept of valence in chemistry. For details
on bases of graph theory, any standard text such as [43] can be of great help.
Cheminformatics is another emerging field in which quantitative structure-activity
and Structure-property relationships predict the biological activities and properties of
nano-material. In these studies, some Physico-chemical properties and topological indices are used to predict bioactivity of the chemical compounds see [5, 9, 26, 45, 39].
Algebraic polynomials have also useful applications in chemistry such as Hosoya polynomial (also called Wiener polynomial) [18] which play a vital role in determining distancebased topological indices. Among other algebraic polynomials, the M-polynomial [10]
introduced in 2015, plays the same role in determining the closed form of many degreebased topological indices [1, 31, 32, 33, 34]. The main advantage of the M-polynomial
is the wealth of information that it contains about degree-based graph invariants.
The line graph L(G) of a graph G is the graph each of whose vertices, represents an
edge of G and two of its vertices are adjacent if their corresponding edges are adjacent
in G. In this article, we compute closed form of some degree-based topological indices
of the line graph of Banana tree graph by using the M-polynomial. Some of these
topological indices were calculated directly in [40].
2.
Basic definitions and literature review
Here we give some basic definitions and literature review.
Some Computational Aspects for the line graph of Banana Tree Graph
3
Definition 2.1. The M-polynomial of G is defined as:
M(G, x, y) =
mij (G)x i y j ,
δ≤i≤j ≤
where δ = min{dv : v ∈ V (G)}, = max{dv : v ∈ V (G)}, and mij (G) the number of
edges vu ∈ E(G) such that {dv , du } = {i, j }.
Weiner [44] in 1947 approximated the boiling point of alkanes as αW (G)+βP3 +γ ,
where α, β and γ are empirical constants, W (G) is the Weiner index and P3 is the number
of paths of length 3 in G. Thus Weiner laid the foundation of topological index which
is also known as connectivity index. A lot of chemical experiments require determining
the chemical properties of emerging nanotubes and nanomaterials. Chemical-based
experiments reveal that out of more than 140 topological indices, no single index is strong
enough to determine many physico-chemical properties, although, in combination, these
topological indices can do this to some extent. The Wiener index is originally the first and
most studied topological index, see for details [11, 21]. Randić index, [36] denoted by
R−1/2 (G) and introduced by Milan Randić in 1975, is also one of the oldest topological
indices. The Randić index is defined as
1
R−1/2 (G) =
.
√
d
d
u
v
uv∈E(G)
In 1998, working independently, Bollobas and Erdos [4] and Amic et al. [2] proposed
the generalized Randić index and has been studied extensively by both chemists and
mathematicians [23] and many mathematical properties of this index have been discussed
in [6]. For a detailed survey we refer the book [27].
1
, and the inverse
The general Randić index is defined as: Rα (G) =
(du dv )α
uv∈E(G)
(du dv )α . Obviously, R−1/2 (G) is the
Randić index is defined as RRα (G) =
uv∈E(G)
1
particular case of Rα (G) when α = − .
2
The Randić index is a most popular, most often applied and most studied index among
all topological indices. Many papers and books such as [24, 25, 27] are written on this
topological index. Randić himself wrote two reviews on his Randić index [37, 38] and
there are three more reviews on it, see [20, 28, 29]. The suitability of the Randić index
for drug design was immediately recognized, and eventually, the index was used for this
purpose on countless occasions. The physical reason for the success of such a simple
graph invariant is still an enigma, although several more-or-less plausible explanations
were offered.
Gutman and Trinajstic [22] introduced
first Zagreb index and second
Zagreb index,
(du + dv ) and M2 (G) =
(du × dv ),
which are defined as: M1 (G) =
uv∈E(G)
uv∈E(G)
respectively. For detail about these indices we refer [8, 19, 35, 41, 42] to the readers.
4
M. S. Ahmad, et al.
Both the first Zagreb index and the second Zagreb index give greater weights to the
inner vertices and edges, and smaller weights to outer vertices and edges which oppose
intuitive reasoning [30]. For a simple connected graph G, the second modified Zagreb
index is defined as:
1
m
M2 (G) =
.
d(u)d(v)
uvE(G)
The symmetric division index [SDD] is the one among 148 discrete Adriatic indices
and is a good predictor of the total surface area for polychlorobiphenyls, see [17]. The
symmetric division index of a connected graph G, is defined as:
SDD(G) =
min(du , dv )
max(du , dv )
+
.
max(du , dv )
min(du , dv )
uv∈E(G)
Another variant of Randić index is the harmonic index defined as
H (G) =
uv∈E(G)
2
.
du + dv
As far as we know, this index first appeared in [14]. Favaron et al. [15] considered the
relation between the harmonic index and the eigenvalues of graphs.
The inverse sum index, is the descriptor that was selected in [3] as a significant
predictor of total surface area of octane isomers and for which the extremal graphs
obtained with the help of mathematical chemistry have a particularly simple and elegant
structure. The inverse sum index is defined as:
I (G) =
uv∈E(G)
du dv
.
du + dv
The augmented Zagreb index of G proposed by Furtula et al. [16] is defined as
A(G) =
uv∈E(G)
du dv
du + dv − 2
3
.
This graph invariant has proven to be a valuable predictive index in the study of heat
of formation in octanes and heptanes (see [16]), whose prediction power is better than
atom-bond connectivity index (please refer to [7, 13, 12] for its research background).
Moreover, the tight upper and lower bounds for the augmented Zagreb index of chemical
trees, and the trees with minimal augmented Zagreb index were obtained in [16].
The following Table 1 relates some well-known degree-based topological indices
Some Computational Aspects for the line graph of Banana Tree Graph
5
with the M-polynomial [10].
Table 1. Derivation of topological indices
topological indices
first Zagreb index
second Zagrab index
modified second Zagrab index
Randić index
inverse Randić index
symmetric division index
harmonic index
Inverse Sum Index
augmented Zagreb index
derivation from M(G; x, y)
(Dx + Dy )(M(G; x, y))x=y=1
(Dx Dy )(M(G; x, y))x=y=1
(Sx Sy )(M(G; x, y))x=y=1
(Dxα Dyα )(M(G; x, y))x=y=1
(Sxα Syα )(M(G; x, y))x=y=1
(Dx Sy + Sx Dy )(M(G; x, y))x=y=1
2Sx J (M(G; x, y))x=y=1
Sx J Dx Dy (M(G; x, y))x=y=1
Sx3 Q−2 J Dx3 Dy3 (M(G; x, y))x=y=1
In Table 1,
∂(f (x, y))
∂(f (x, y))
, Dy = y
,
∂x
∂y
y
x f (t, y)
f (t, y)
dt, Sy =
dt,
t
t
0
0
J (f (x, y)) = f (x, x), Qα (f (x, y)) = x α f (x, y).
Dx = x
Sx =
The following lemmas [19] will be helpful for our results.
Lemma 2.2. Let G be a graph with u, v ∈ V (G) and e = uv ∈ E(G). Then
de = du + dv − 2.
Lemma 2.3. Let G be a graph of order p and size q. Then the line graph L(G) of G is
1
a graph of order p and size M1 (G) − q.
2
3.
Results and discussions
In this part, we give or main computational results. The Banana tree graph Bn,k is the
graph obtained by connecting one leaf of each of n copies of a k-star graph with a single
root vertex that is distinct for all the stars. The Bn,k has order nk + 1 and size nk. B3,5
is shown in Figure 1.
Theorem 3.1. Let G be the line graph of Banana graph. Then the M-polynomial of G
is
n(n − 1) n n
x y + nk k−1 y n + ((k − 2)n)x k−1 y k−2
M(G; x, y) =
2
nk 2 + 6n − 5kn k−2 k−2
+
x y .
2
6
M. S. Ahmad, et al.
Figure 1: The Banana tree graph B3,5
Figure 2: The line graph of Banana tree graph B3,5
Proof. The graph G for n = 3 and k = 5 is shown in Figure 2.
By using Lemma 2.2, it is easy to see that the order of G is nk out of which (k − 2)n
vertices are of degree k − 2, n vertices are of degree k − 1 and n vertices are of degree n.
n62 + 3n + nk 2 − 3nk
. There are four types
Therefore, by using Lemma 2.3, G has size
2
of edges in G based on degrees of end vertices of each edge. The first edge partitions
n(n − 1)
E1 (G), contains
edges uv, where du = dv = n. The second edge partitions
2
E2 (G), contains n edges uv, where du = k −1, dv = n. The third edge partitions E3 (G),
contains (k − 2)n edges uv, where du = k − 1, dv = k − 2 and the forth edge partitions
nk 2 + 6n − 5kn
edges uv, where du = dv = k −2. We take k −1 ≤ n,
E4 (G), contains
2
from Definition 2.1, of the M-polynomial of G, we have
M(G; x, y) =
mij x i y j
i≤j
=
m≤n
+
mnn x n y n +
k−1≤k−2
m(k−1)n x k−1 y n
k−1≤n
m(k−1)(k−2) x k−1 y k−2 +
k−2≤k−2
m(k−2)(k−2) x k−2 y k−2
Some Computational Aspects for the line graph of Banana Tree Graph
=
uv∈E1 (G)
+
mnn x n y n +
7
m(k−1)n x k−1 y n
uv∈E2 (G)
m(k−1)(k−2) x k−1 y k−2 +
m(k−2)(k−2) x k−2 y k−2
uv∈E3 (G)
uv∈E4 (G)
n n
k−1 n
= |E1 (G)|x y + |E2 (G)|m(k−1)n x y + |E3 (G)|x k−1 y k−2
+ |E4 (G)|x k−2 y k−2
=
n(n − 1) n n
x y + nx k−1 y n + ((k − 2)n)x k−1 y k−2
2
nk 2 + 6n − 5kn k−2 k−2
x y .
+
2
Figure 3: Plot of M-polynomial for the line graph of Banana graph
Next we compute some degree-based topological indices of the line graph of banana
tree from this M-polynomial.
Corollary 3.2. Let G be the line graph of the Banana tree. Then
1. M1 (G) = n(k 3 − 5k 2 + (n + 10)k + n2 − 2n − 7).
1
2. M2 (G) = (k 4 − 7k 3 + 20k 2 + (2n − 28)k + n3 − n2 − 2n + 16)n.
2
3.
m
(n2 + n − 1)k 2 + (−2n2 − n + 3)k − n2 − 2n − 2
.
M2 (G) =
2(k − 2)(k − 1)n
8
M. S. Ahmad, et al.
1
1
4. Rα (G) = n2α+1 (n−1)+ n(k −2)(k −3)(k −2)2α +((k −2)α+1 n+nα+1 )(k −
2
2
1)α .
1 n(n − 1)
n
n(k − 2)
n(k 2 − 5k + 6)
5. RRα (G) =
+
+
+
.
2 n2α
(k − 1)α nα (k − 1)α (k − 2)2α
2(k − 2)2α
6. SSD(G) =
7. H (G) =
1 + k 3 n + (−4n + 1)k 2 + (n2 + 4n − 2)k
.
(k − 1)
n−1
n
n(k − 2) n(k 2 − 5k + 6)
+
.
+
+
2k − 3
4k − 8
16
n+k−1
1 2
2(k − 1)n n(k − 2)2 (k − 1) n(k − 2)3 (k − 3)
8. I (G) = n (n − 1) +
+
+
.
n+k−1
2k − 3
4k − 8
4
9. A(G) =
n7
(k − 1)7 n4
n(k − 2)4 (k − 1)3
n(k − 2)7
+
+
+
.
16(n − 1)2 (n + k − 3)3
(2k − 5)3
16(k − 3)2
Proof. Let
M(G; x, y) = f (x, y)
n(n − 1) n n
x y + nx k−1 y n + ((k − 2)n)x k−1 y k−2
=
2
nk 2 + 6n − 5kn k−2 k−2
x y .
+
2
Then
Dx f (x, y) =
n2 (n − 1) n n
x y + n(k − 1)x k−1 y n
2
+ ((k − 1)(k − 2)n)x k−1 y k−2 + (k − 2)
nk 2 + 6n − 5kn k−2 k−2
x y ,
2
n2 (n − 1) n n
x y + n2 x k−1 y n + ((k − 2)2 n)x k−1 y k−2
2
nk 2 + 6n − 5kn k−2 k−2
+ (k − 2)
x y ,
2
n3 (n − 1) n n
Dy Dx f (x, y) =
x y + (k − 1)n2 x k−1 y n + ((k − 1)(k − 2)2 n)x k−1 y k−2
2
2
2 nk + 6n − 5kn k−2 k−2
+ (k − 2)
x y ,
2
n(n − 1) n n
x y + x k−1 y n + nx k−1 y k−2
Sy (f (x, y)) =
2n
nk 2 + 6n − 5kn k−2 k−2
x y ,
+
2
Dy f (x, y) =
Some Computational Aspects for the line graph of Banana Tree Graph
Sx Sy (f (x, y)) =
Dyα (f (x, y)) =
n
1
(n − 1) n n
x k−1 y k−2
x k−1 y n +
x y +
k−1
k−1
2n
nk 2 + 6n − 5kn k−2 k−2
+
x y ,
2(k − 2)2
nα+1 (n − 1) n n
x y + nα+1 x k−1 y n + ((k − 2)α+1 n)x k−1 y k−2
2
nk 2 + 6n − 5kn k−2 k−2
x y ,
+ (k − 2)α
2
Dxα Dyα (f (x, y)) =
n2α+1 (n − 1) n n
x y + (k − 1)α nα+1 x k−1 y n
2
+ ((k − 1)α (k − 2)α+1 n)x k−1 y k−2
nk 2 + 6n − 5kn k−2 k−2
x y ,
2
n(n − 1) n n
n
(k − 2)n k−1 k−2
x y + α x k−1 y n +
x y
Syα (f (x, y)) =
α
2n
n
(k − 2)α
nk 2 + 6n − 5kn k−2 k−2
+
x y ,
2(k − 2)α
+ (k − 2)2α
Sxα Syα (f (x, y)) =
n
n(n − 1) n n
x k−1 y n
x y +
2α
(k − 1)α nα
2n
(k − 2)n
nk 2 + 6n − 5kn k−2 k−2
k−1 k−2
+
x
y
+
x y ,
(k − 1)α (k − 2)α
2(k − 2)2α
Sy Dx (f (x, y)) =
n(n − 1) n n
x y + (k − 1)x k−1 y n + (k − 1)nx k−1 y k−2
2
nk 2 + 6n − 5kn k−2 k−2
+
x y ,
2
n(n − 1) n n
n2 k−1 n (k − 2)2 n k−1 k−2
x y +
x y +
x y
2
k−1
k−1
nk 2 + 6n − 5kn k−2 k−2
+
x y ,
2
n(n − 1) 2n
x + nx n+k−1 + (k − 2)nx 2k−3
Jf (x, y) =
2
nk 2 + 6n − 5kn 2k−4
x
+
,
2
(n − 1) 2n
n
(k − 2)n 2k−3
x +
x n+k−1 +
x
Sx Jf (x, y) =
4
n+k−1
2k − 3
nk 2 + 6n − 5kn 2k−4
x
+
,
4k − 8
Sx Dy (f (x, y)) =
9
10
M. S. Ahmad, et al.
n3 (n − 1) 2n
x + (k − 1)n2 x n+k−1 + (k − 1)(k − 2)2 nx 2k−3
2
nk 2 + 6n − 5kn 2k−4
x
+ (k − 2)2
,
2
n2 (n − 1) 2n (k − 1)n2 n+k−1 (k − 1)(k − 2)2 n 2k−3
Sx J Dx Dy f (x, y) =
x +
x
x
+
4
n+k−1
2k − 3
nk 2 + 6n − 5kn 2k−4
,
x
+ (k − 2)2
4k − 8
J Dx Dy f (x, y) =
n4 (n − 1) n n
x y + n4 x k−1 y n + (k − 2)4 nx k−1 y k−2
2
nk 2 + 6n − 5kn k−2 k−2
x y ,
+ (k − 2)4
2
n7 (n − 1) n n
Dx3 Dy3 f (x, y) =
x y + (k − 1)7 n4 x k−1 y n + (k − 1)3 (k − 2)4 nx k−1 y k−2
2
2
6 nk + 6n − 5kn k−2 k−2
+ (k − 2)
x y ,
2
n7 (n − 1) 2n
J Dx3 Dy3 f (x, y) =
x + (k − 1)7 n4 x n+k−1 + (k − 1)3 (k − 2)4 nx 2k−3
2
nk 2 + 6n − 5kn 2k−4
x
+ (k − 2)6
,
2
n7 (n − 1) 2n−2
Q−2 J Dx3 Dy3 f (x, y) =
x
+ (k − 1)7 n4 x n+k−3
2
nk 2 + 6n − 5kn 2k−6
+ (k − 1)3 (k − 2)4 nx 2k−5 + (k − 2)6
,
x
2
n7 (n − 1) 2n−2
(k − 1)7 n4 n+k−3
Sx3 Q−2 J Dx3 Dy3 f (x, y) =
x
+
x
2(2n − 2)3
(n + k − 3)3
2
(k − 1)3 (k − 2)4 n 2k−5
6 nk + 6n − 5kn 2k−6
+
x
+
(k
−
2)
x
.
(2k − 5)3
2(2k − 6)3
Using Table 1, we have the following graphs of different indices.
Dy3 f (x, y) =
1.
M1 (G) = (Dx + Dy )f (x, y)|x=y=1
= n(k 3 − 5k 2 + (n + 10)k + n2 − 2n − 7).
2.
M2 (G) = Dy Dx (f (x, y))|x=y=1
1
= (k 4 − 7k 3 + 20k 2 + (2n − 28)k + n3 − n2 − 2n + 16)n.
2
Some Computational Aspects for the line graph of Banana Tree Graph
11
Figure 4: Plot for the first Zagreb index for the line graph of Banana tree graph.
Figure 5: Plot for the first Zagreb index for the line graph of Banana tree graph for k = 1.
3.
m
M2 (G) = Sx Sy (f (x, y))|x=y=1
=
(n2 + n − 1)k 2 + (−2n2 − n + 3)k − n2 − 2n − 2
.
2(k − 2)(k − 1)n
12
M. S. Ahmad, et al.
Figure 6: Plot for the first Zagreb index for the line graph of Banana tree graph for n = 1.
Figure 7: Plot for the second Zagreb index for the line graph of Banana tree graph.
4.
Rα (G) = Dxα Dyα (f (x, y))|x=y=1
1
1
= n2α+1 (n − 1) + n(k − 2)(k − 3)(k − 2)2α
2
2
α+1
+ ((k − 2) n + nα+1 (k − 1)α .
Some Computational Aspects for the line graph of Banana Tree Graph
13
Figure 8: Plot for the second Zagreb index for the line graph of Banana tree graph for
k = 1.
Figure 9: Plot for the second Zagreb index for the line graph of Banana tree graph for
n = 1.
5.
RRα (G) = Sxα Syα (f (x, y))|x=y=1
1 n(n − 1)
n
n(k − 2)
n(k 2 − 5k + 6)
=
+
+
+
.
2 n2α
(k − 1)α nα (k − 1)α (k − 2)α
2(k − 2)2α
14
M. S. Ahmad, et al.
Figure 10: Plot for the modified second Zagreb index for the line graph of Banana tree
graph.
Figure 11: Plot for the modified second Zagreb index for the line graph of Banana tree
graph for k = 3.
6.
SSD(G)(Sy Dx + Sx Dy )(f (x, y))|x=y=1
1 + k 3 n + (−4n + 1)k 2 + (n2 + 4n − 2)k
=
.
(k − 1)
Some Computational Aspects for the line graph of Banana Tree Graph
15
Figure 12: Plot for the modified second Zagreb index for the line graph of Banana tree
graph for n = 1.
Figure 13: Plot for the generalized Randić index for the line graph of Banana tree graph
1
for α = .
2
7.
H (G) = 2Sx J (f (x, y))x=1
=
n
n(k − 2) n(k 2 − 5k + 6)
n−1
+
+
+
.
16
n+k−1
2k − 3
4k − 8
16
M. S. Ahmad, et al.
Figure 14: Plot for the generalized Randić index for the line graph of Banana tree graph
1
for k = 4 and α = .
2
Figure 15: Plot for the generalized Randić index for the line graph of Banana tree graph
1
for n = 1 and α = .
2
8.
I (G) = Sx J Dx Dy (f (x, y))x=1
1
2(k − 1)n n(k − 2)2 (k − 1)
= n2 (n − 1) +
+
4
n+k−1
2k − 3
3
n(k − 2) (k − 3)
.
+
4k − 8
Some Computational Aspects for the line graph of Banana Tree Graph
17
Figure 16: Plot for the inverse Randić index for the line graph of Banana tree graph for
1
α= .
2
Figure 17: Plot for the inverse Randić index for the line graph of Banana tree graph for
1
k = 3 and α = .
2
18
M. S. Ahmad, et al.
Figure 18: Plot for the inverse Randić index for the line graph of Banana tree graph for
1
n = 1 and α = .
2
Figure 19: Plot for the symmetric division index for the line graph of Banana tree graph.
Some Computational Aspects for the line graph of Banana Tree Graph
19
Figure 20: Plot for the symmetric division index for the line graph of Banana tree graph
for k = 2.
Figure 21: Plot for the symmetric division index for the line graph of Banana tree graph
for n = 1.
20
M. S. Ahmad, et al.
Figure 22: Plot for the harmonic index for the line graph of Banana tree graph.
Figure 23: Plot for the harmonic index for the line graph of Banana tree graph for k = 1.
Some Computational Aspects for the line graph of Banana Tree Graph
21
Figure 24: Plot for the harmonic index for the line graph of Banana tree graph for n = 1.
Figure 25: Plot for the inverse sum index for the line graph of Banana tree graph.
22
M. S. Ahmad, et al.
Figure 26: Plot for the inverse sum index for the line graph of Banana tree graph for
k = 4.
Figure 27: Plot for the inverse sum index for the line graph of Banana tree graph for
n = 1.
Some Computational Aspects for the line graph of Banana Tree Graph
23
Figure 28: Plot for the augmented Zagreb index for the line graph of Banana tree graph.
Figure 29: Plot for the augmented Zagreb index for the line graph of Banana tree graph
for k = 4.
24
M. S. Ahmad, et al.
9.
A(G) = Sx3 Q−2 J Dx3 Dy3 (f (x, y))
=
n7
(k − 1)7 n4
n(k − 2)4 (k − 1)3
+
+
(2k − 5)3
16(n − 1)2 (n + k − 3)3
n(k − 2)7
.
+
16(k − 3)2
Figure 30: Plot for the augmented Zagreb index for the line graph of Banana tree graph
for n = 2.
4.
Conclusions
In this article we compute many topological indices for line graph of banana tree. At
first we give general closed forms of M-polynomial of this graph and then recover
many degree-based topological indices out of it. These results can play a vital rule in
preparation of new drugs.
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