MATCH.
COlllmunications in Mathematical
and in Computer Chemist/y
MATCH Commun. Matho Computo Chemo 57 (2007) 183-191
ISSN 0340 - 6253
ON SPECTRAL MOMENTS AND ENERGY OF
GRAPHS
Bo Zhou,a Ivan Gutman,b José Antonio de la Peña,c
Juan Radad and Leonel Mendozad
a
Department 01 Mathematics, South China Normal University,
Guangzhou 510631, Po R. China
e-mail: zhoubo@senu edu en
o
o
bFaculty 01 SC'ience, University 01 Kmgujevac,
Po 00 Box 60, 34000 Kmgujevac, Serbia
e-mail:
gutman@kg
oae oyu
eInstituto de Matemáticas, UNA M, Circuito Exterior,
Ciudad Universitaria, México 04510, Do Fo, México
e-mail: j ap@penelope matem unam mx
o
d Departamento
o
o
de Matemática, Univer"sidad de Los Andes
Mérida 5101, Venezuela
e-mail: juanrada@ulao ve
(Received December 17, 2005)
Abstract
Let G be a graph on n vertices, alld let Al, A2,..., An be its cigcnvalues. Thc encrgy of
Gis E
= L:i=lIAiloThc k-th spectral moment of Gis
Nh = L:i=l(A;)k We prove that for
o
cvcn positive integcrs 10,s, t , such that 4r = s + t + 2, thc inequality E ~ (Mr)2 /..¡ Ms Mt
holds for all graphs with at least one edge, thus generalizing an earlier result [de la PeIia,
MClldoza,Rada, DisC1.0
Math. 302 (2005)77-84]. The graphs for which thc aboverelation
bCCOIllCS
cquality are charactcrizedo
- 184 -
INTRODUCTION
Let G be a graph without loops and l11ultipleedges. Let n and m be, respectively, the number of vertices and edges of G. The eigenvalues of G are denoted by
Al, A2, . . . , An , ancl are assumed to be labelled in a non- increasing manner:
Al
~ A2 ~ . . . ~ An .
The basic properties of graph eigenvalues can be founel in the book [1].
The energy of the graph G is defined as
T1
E = E(G) = ¿
IAil
i=1
The graph-energy
(1)
.
concept has a chemical motivation.
Namely, for graphs which
in the Hückel molecular orbital theory represent the carbon-atom
conjugated
hydrocarbons,
skeleton of some
E is related to the total 7r-electron energy (ormore
pre-
cisely: the total 7r-electron energy is a linear function of E). For more details on this
matter see the book [2] ancI the recent review [3].
However, more than a quarter of century ago, one of the present authors proposed
[4] that the graph invariant E irrespective
as defined above -
of its possible chemicaI interpretation.
be considered for all graphs,
This change of viewpoint made
it possible to envisage and establish numerous new, generall)' valid, mathematical
properties
of E, some of evident. value for chemistry, so me of no visible chemical
applicabili ty.
After 1978 the graph-energy concept was presented to the mathematieo--chemical
community on several othcr oecasions [2, 5, 6]. Initially, the response of other eolleagues was almost ni!. However, around the turn of the century the study of E suddenly became a rather popular topie both in mathematieal chemistry and in "pure"
mathematics. Of the numerous papers on graph energy that recently appeared, we
quote here only a few [7-29].
For a nonnegative integer k, the k-th spectml rnoment of the graph G is defined
as
n
l1h
= Ah(G) = ¿(Ail .
i=1
(2)
- 185 -
Note that Mk is equal to the nurnber of closee!walks of length k in G [1].
B€cause both the energy ane! the spectral moments are symmetric functions of
graph eigenvalues, it is reasonable to ask if there exist relations between them. Much
work has been devoted to problems of this kind [30-39].
Rccently, de la Peña et al. [25] proved the following:
Theorem
1. Let G be a bipartite graph with at least ane cdge and let l' ,
s, t be even
pasitive integers, such that 41'= s + t + 2. Then
(3)
E(G) 2: M,.(Gf [Ms(G) Mt(G)¡-1/2 .
In this article we show that the statement
of Theorem
1 can be, iu a natural
mauner, extended to all graphs, and also offer a simple proof of the inequality Eq.
(3).
EXTENDING
THEOREM
1
In Eq. (3) it is essential that thc parameters s and t are (non-ncgative) even intcgers, because in the case of bipartite graphs, the odd spectral moments are llecessarily
zcro [1]. In arder to overcome this lirnitation \Ve define the rnornent-like
quantities
n
M; = M;(G) = ¿ 1\lk .
(4)
i=l
Rere k may be an odel illteger, but also any real-valued number. Comparing Eqs. (2)
"
and (4) we concluele that if k is an even integer, then Mi. = Mk. Bearing in mind
(1), we see that Ají = E.
The main result of this section is the following extension of Theorem 1:
Theorem
la. Let G be o.graph with at least ane edge and let l' , S , t be nan-negative
real nv.mbers, such that 4r
= s + t + 2. Then
E(G) 2: M;(G)2 [M;(G) .;Vl;(G)¡-1/2 .
(5)
Evidently, Theorem 1 is a special case of Theorern la for G being a hipartite graph
ane! for sane! t being even positive integers.
- ] 86 -
In order tú prove Theorem la we need a simple' lemma.
Lemma
2.
non-negative
Lct (/1,0.2.. . . , ah be positive real nll,mbe1's, h > 1, o.nd let T, s, t be
real n-urnbe1's,s'Uch that 4T = s + t + 2. Then
,1
J¡
[f;(aJ
1f (s, t) =1(1,1),
]
2
J¡
(
S
f; )
h
h
(6)
Cl¡
f;(aJ'
{;(a¡)t
.
then the eq'Uality in (6) holds if o.nd only if al
= 0.2 = . .. = ah .
Note that if s = t = 1 (amI, consequently, T = 1), then equality in (6) holds in a
trivial mannel' fol' auy choice of a;'s.
Proof.
By the Cauchy-Schwarz
4
h
[ ~(aJ
]
inequality,
4
h
=
[ ~(a¡)(S+t)/4.
h
2
h
(a¡)1/2] S [ (;(a¡)(S+1)/2r; a¡]
2
h
h
2
h
h
2
h
= [ t;(a¡)S/2. (a¡)t/2
]
[ f; a¡] S {;(air {;(ad ( r; a¡)
This proves (6). Equality in (6) holds if and only if both 0.:+1-2and o.:-t are constant
for al! i = 1,2,..., h. Thus if (s, t) =1(1,1), then equality in (6) holds if and only if
al
= 0.2 = . . . = ah .
Proof
gl'aph
of Theorem
G.
o
la.
Let /l1
~
/l2
~ . .. ~ /lh be the non-zero eigenvalues of the
Sincc G has at least one edge, we have [1] /l1
=
Al > O and /lh
=
An
<
O.
Using Lemma 2 fol' the positive numbcl's a¡ = IILil, i~= 1,2,..., h, and noting
h
h
that in this case L (ai)k = M¡(G) ando in particular, L ai = E(G), we have
i=li=l
M,~(G)'" S E(G)2 M;(G) M;(G)
from which (5) follows.
If s
= t = 1, then
(s, t) =1 (1,1),
o
in a trivial manner (5) becomes an equality. By Lemma 2, if
then equality in (5) holds if and ol1ly if Ift11=
now determine the graphs that obey these conditions.
1/l21
= . .. =
l/lhl.
\Ve
- ] 87 -
Theorem
3. Equality in (5) holds if and only if the cornponents of the graph G are
isolated vert-ices and/o1' complete biparlite
gmphs J(Pl,qp
. .. ,J(Pk.qk fo1' sorne k 2: 1,
such that p¡ q¡ = . . . = Pk qk .
Proof.
e fori = 1,. . ., k,
If G is a graph specified in the theorem with Piqi =
easy to check [11that its non-2ero eigenvalues are
it is
VC (k t,imes)and -VC (k times),
and so the equality in (5) holds.
Suppose that tbe equality in (5) holds. Then IfLd= IfL21= ... = IfLhl. Note
h
that,
by Theorem 0.13 in [1], it must be
G must be symmetric
is a bipartite
graph.
L fLi =
.i=¡
O. Therefore
the spectrum
of
w. r. t. the origin, and thus (by Theorem 3.11 in [1]), G
Note also that G has either two (fL¡ and -fL¡) or three (fL¡,
- fL¡, and O) distinct eigenvalues.
Then by Theorems 6.4 and 6.5 in [1], G is the
disjoint union of complete hipartite
graphs Kp¡.lJlI' . . ,Kpbqk for some k 2: 1, such
that Pl q¡ = . . . = Pkqk, and possibly isolated vertices.
o
DISCUSSION
By setting (s, t) = (0,2) in Theorem la (which implies
v2m(n
l'
= 1), we obtain E S;
- no) where no is the number of zero eigenvalues of the unclerlying graph.
This impro\'ement
of the famous McClelland upper bound E S; y2mn
[2, 31 seems
to be first reported in [40].
By setting (s,t) = (2,4) in Theorem la (whichimpliesl' = 2), we obtain the lower
bound
E
E ~ 2V2m
V --¡;¡¡
.
For bipartite graphs it has been reported in [13, 25], whereas for all (both bipartite
and non-bipartite) graphs in [16, 41].
* * * * *
The relation 41' = s + t + 2 usecl in Lemma 2 may be viewed as a special case of
2k l' =
S
+ t + 2k - 2
(7)
- 188-
fol' k
= 2. With condition (7), with k ?: 2 and k integel', instead of (6) we get
h
2'
S
[~(air]
2' - 2
h
(t;ai)
h
h
t;(ai)S ~(ad
.
which, in turn, implies
E(G) ?: M;(Gt/(2k_2) [M;(G) .M;(G)¡-1/(2'-2) .
The condition
41'
=
s + t + 2 can be furthcr modified. For instancc, if 81' =
s + t + u + 5, then
8
h
[ ~(air ]
h
5
h
h
h
) ~(a¡)S{;(ad {;(a.it
S ( ~ Q.¡
and
E(G) ?: M,~(G)~/5 [M;(G) l'v1;(G) MZ(G)¡-1/5
If 81' = s +
t+
ti
.
+ v + 4, then
8
h
[ ~(air ]
h
4
h
h
h
h
(
S {;ai ) {;(air {;(a;)t{;(ait t;(a;)V
and
E(G) ?: M; (G)2 [M;(G) M;(G) MZ(G) lvl:(G)¡-1/4 .
Acknowledgerncnt: This work was supported by the Guangdong Provincial Natural
Science Foundation of China (no. 05005928), by the Serbian tvIinistry of Science
ancl Environmental Protection, through Grant no. 144015G, ancl by CDCHT-ULA,
Projcct No. C-13490505B.
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