Journal of Chemical and Pharmaceutical Research, 2014, 6(3):477481
Research Article
ISSN : 09757384
CODEN(USA) : JCPRC5
On the extremal hyperwiener index of graphs
Li Yan
1_{, YingfangLi}
1_{, Wei Gao}
2_{and Junsheng Li}
11
_{School of Engineering, Honghe University, Mengzi, P. R. China}
2
_{School of Information Science and Technology, Yunnan Normal University, Kunming, P. R. China}
____________________________________________________________________________________________
ABSTRACT
In this paper, we consider the relationship between HyperWiener index and some special parameters of graph, and present the graphs which minimize the HyperWiener index among all graphs with given chromatic number and clique number and the graphs which maximum the HyperWiener index among all graphs with given chromatic number and clique number.
Keywords:Chemical graph theory, organic molecules, HyperWiener index, chromatic number, clique number ____________________________________________________________________________________________
INTRODUCTION
The HyperWiener index, as an extension of Wiener index, is an important topological index in Chemistry. It is used for the structure of molecule. There is a very close relation between the physical, chemical characteristics of many compounds and the topological structure of that. The HyperWiener index is such a topological index and it has been widely used in Chemistry fields. Some conclusion for HyperWiener index can refer to [1].
The graphs considered in this paper are simple and connected. The vertex and edge sets ofGare denoted byV(G) andE(G), respectively. The Wiener index is defined as the sum of distances between all unordered pair of vertices of a graphG, i.e.,
( )
W G
={ , } ( )
( , )
u v V G
d u v
,where
d u v
( , )
is the distance betweenuandvinG.Several papers contributed to determine the Wiener index of special graphs. Gao and Shi [2] determined the Wiener index of gear fan graph, gear wheel graph and theirrcorona graphs. Chen [3]gained the exact expression for general pepoid graph. Xing and Cai [4] characterized the tree with thirdminimum wiener index and introduce the method of obtaining the order of the Wiener indices among all the trees with given order and diameter, respectively. A tricyclic graph is a connected graph withn vertices andn+2 edges. Wan and Ren [5] studied the Wiener index of tricyclic graph
_{n}3 which have at most a common vertex between any two circuits, and the smallest, the secondsmallestWiener indices of the tricyclic graphs
n3 are given. The HyperWiener index WW is one of the recently distancebased graph invariants. That WW clearly encodes the compactness of a structure and the WW of G is define as:( )
WW G
= 2{ , } ( ) { , } ( )
1
(
( , )
( , ))
2
u v V G u v V Gd u v
d u v
Pan [6] deduced the formula of Wiener number and HyperWiener number of two types of polyomino systems. More results on Wiener index and HyperWiener index can refer to [714].
In this paper, we connect the HyperWiener index with some wellknown graph theoretic parameters, such as chromatic number and clique number. It is proved that among all graphs withnvertices and chromatic numberk, the HyperWiener index is minimized by the graph
T
_{k n}_{,} , and among all graphs withnvertices and clique numberk, theHyperWiener index is minimized by the graph
T
_{k n}_{,} and maximized by the graphK P
_{k}
_{n k}_{} . Note thatT
_{k n}_{,} is completekpartite graph onn vertices in which each part has either
_{}
n k
/
_{}
or
_{}
n k
/
_{}
vertices. Let
( )
G
andc(G) be the chromatic number and clique number of graphG, respectively.1. Main results and proof
Theorem 1.LetGbe a graph onnvertices and with
( )
G
=k. Lett=
_{}
n k
/
_{}
. Then( )
WW G
3 (
1)
2
n n

2
2
n t
1
2(
1)
2
t
k
_{ }
_{}
(1)and the equality holds if and only ifG
T
_{k n}_{,} .Proof.LetG* be a graph with minimum HyperWiener index in all graphs withnvertices and chromatic numberk. Then the vertex set ofG* can be divided intokparts such that no edges joins two vertices from same part. Moreover,
G* includes all edges joining vertices in different parts. Otherwise, there exists two disconnect vertices vand v’ which belong to different parts. Then the graphG*+vv’ has chromatic numberkand fewer HyperWiener index than
G*, which contradicts to the selection of G*. Hence, G* is a complete kpartite graph
K r r
_{k}( , , , )
_{1} _{2}
r
_{k} withr1+r2+…+rk=n,whereriis the number of vertex inith part (1
i
k).We now claim thatG*
T
_{k n}_{,} . Otherwise, the parts are not as equal as possible, suppose there arerivertices in theith part and
r
j
r
i2
in the jth part. Then by transferring one vertex from the jth part to the ith part, theHyperWiener index will decrease which contradicts to the selection ofG*.
Since
E T
(
k n_{,})
=2
n t
+1
(
1)
2
t
k
_{ }
_{}
. We have,
(
k n)
WW T
=, ,
2
{ , } ( ) { , } ( )
1
(
( , )
( , ))
2
u v V T_{k n} u v V T_{k n}d u v
d u v
=
1
{(4
3[
(
1)
1
]) (2
[
(
1)
1
])}
2
2
2
2
2
2
2
n
n t
t
n
n t
t
k
k
=
3 (
1)
2
n n

2
2
n t
1
2(
1)
2
t
k
_{ }
_{}
.The proof above implies that equality holds in (1) if and only if G
T
_{k n}_{,} . The Theorem thus follows.
Our second result depends heavily on the following lemma.
Lemma 1. [15] Let G be a graph on n vertices. IfG contains on Km+1, then
E G
( )
E T
(
_{k n}_{,})
. Moreover,( )
E G
=E T
(
k n,)
only ifG
T
k n, .following theorem, we only consider the graphGonnvertices with clique numberc(G)<n1. Let
d G i
( , )
be the number of vertex pairs at distancei. LetK P
_{k}
_{n k}_{} be the graph obtained fromK
_{k} andP
_{n k}_{} by joining a vertex ofK
_{k} to one end vertex ofP
_{n k}_{} .Theorem 2.LetGbe a graph onnvertices with clique numberc(G)=k<n1. Then, we have
3 (
1)
2
n n

2
2
n t
1
2(
1)
2
t
k
_{ }
_{}
WW G
( )
2 1 21 2
1
((
(
1)
(
1)
)
2
2
n k n k
i i
k
n k i
i
k
i
1
(
1)(
2)
(
)(
1)
(
(
1)[
1]
))
2
3
2
2
k
n k
n k
n k
n k n k
k
_{ }
_{}
wheret=
_{}
n k
/
_{}
. Moreover, the lower bound is achieved if and only ifG
T
_{k n}_{,} and the upper bound is achieved if and only ifG
K P
_{k}
_{n k} .Proof.LetGbe a graph onnvertices with clique numberc(G)=k<n1 andlbe the diameter ofG. Then, in terms of Lemma 1, we get
( )
WW G
= 2{ , } ( ) { , } ( )
1
(
( , )
( , ))
2
u v V G u v V Gd u v
d u v
= 2 2 21
(( ( )
( , )) ( ( )
( , )))
2
l l i iE G
i d G i
E G
id G i
2 2
1
(( ( ) 4
( , )) ( ( ) 2
( , )))
2
l l
i i
E G
d G i
E G
d G i
(2)=
1
(( ( ) 4[
( ) ]) ( ( ) 2[
( ) ]))
2
2
2
n
n
E G
_{ }
E G
E G
_{ }
E G
=
3
2 ( )
2
n
E G
3 (
1)
2 (
_{,})
2
k nn n
E T
_{}
(3)
=
3 (
1)
2
n n

2
2
n t
1
2(
1)
2
t
k
_{}
.It is fact that equality in both (2) and (3) will hold if and only ifG
T
k n, . Note that the clique number of graph,
k n
T
isk. Hence, we obtain( )
WW G
3 (
1)
2
n n

2
2
n t
1
2(
1)
2
t
k
_{ }
_{}
,and the equality holds if and only ifG
T
_{k n}_{,} .In the following, we will prove the upper bound on HyperWiener index by induction on vertex numbern.
Clearly,G* has a pendent vertex, sayu. Letvbe the pendent vertex of
K P
_{k}
_{n k}_{} . Then, we infer( *)
WW G
=WW G
( *
u
)
+d u
_{G}_{*}( )
,(
_{k} _{n k})
WW K P
_{} =WW K P
(
_{k}
_{n k}_{}
v
)
+d
K P_{k}_{n k}( )
v
.Obviously, G*u has n1 vertices and with clique number k. In view of induction hypothesis, we deduce
( *
)
WW G
u
WW K P
(
_{k}
_{n k}_{}
v
)
. Sinced u
_{G}_{*}( )
d
K P_{k}_{n k}( )
v
and the equality holds if and only ifG
K P
_{k}
_{n k}_{} . We yieldWW G
( *)
WW K P
(
_{k}
_{n k})
.By straightforward calculation, we have
(
_{k} _{n k})
WW K P
_{} = 2{ , } ( ) { , } ( )
1
(
( , )
( , ))
2
u v V K P_{k} _{n k} u v V K P_{k} _{n k}d u v
d u v
=
1
2 2
1 2
1
((
(
1)
(
1)
)
2
2
n k n k
i i
k
n k i
i
k
i
1
(
1)(
2)
(
)(
1)
(
(
1)[
1]
))
2
3
2
2
k
n k
n k
n k
n k n k
k
_{ }
_{}
.Thus,
( )
WW G
2 1 21 2
1
((
(
1)
(
1)
)
2
2
n k n k
i i
k
n k i
i
k
i
_{}
_{ }
_{ }
1
(
1)(
2)
(
)(
1)
(
(
1)[
1]
))
2
3
2
2
k
n k
n k
n k
n k n k
k
_{ }
_{}
.and the proof above reveals that the equality holds if and only ifG
K P
_{k}
_{n k}_{} .
CONCLUSION
The contributions of our paper are determining the upper bound and lower bound of HyperWiener index under fixed chromatic number and clique number. Our results also present the sufficient and necessary condition for reaching the upper and lower bound.
Acknowledgements
First, we thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by pecuniary aid of Yunnan Province basic research for application(2013fz127).We also would like to thank the anonymous referees for providing us with constructive comments and suggestions.
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