New Bounds for Zagreb Eccentricity Indices

The Zagreb eccentricity indices are the eccentricity version of the classical Zagreb indices. The first Zagreb eccentricity index (E1(G)) is defined as sum of squares of the eccentricities of the vertices and the second Zagreb eccentricity index (E2(G)) is equal to sum of product of the eccentricities of the adjacent vertices. In this paper we give some new upper and lower bounds for first and second Zagreb eccentricity indices.


Introduction
Let G be a simple connected graph with vertex set V(G) and edge set E(G) so that For any vertex , let deg(v) denote the degree of the vertex v.For vertices u and , the distance between u and v is defined as length of the shortest path connecting u and v and is denoted by d(u,v).The eccentricity of a vertex , denoted by , is the distance between v and a vertex farthest from v i.e.
. The radius and diameter of a graph is the minimum and maximum eccentricity among the vertices of G i.e.
respectively.Also the total eccentricity of a graph, denoted by The first and second Zagreb index of a graph were first introduced by Gutman in [2] which are the most known and widely used topological indices, defined as respectively Recently several Graph invariants based on vertex eccentricities subject to large number of studies.Analogues to Zagreb indices M. Ghorbani et al. [3] and D. Vukiče-vić et al. [4] defined the Zagreb eccentricity indices by replacing degrees by eccentricity of the vertices.Thus the first and second Zagreb eccentricity indices of a graph G are defined as The lower and upper bounds of n-vertex trees with fixed diameter and matching number and extremal trees with respect to Zagreb eccentricity indices were studied by R. Xing et al. [5] and recently K. C. Das et al. in [6] presented some properties, upper and lower bounds of Zagreb eccentricity indices and also characterize the extremal graphs.
Another useful eccentricity and degree based topological index called eccentric connectivity index was first introduced by Sharma, Goswami and Madan [7] and is defined as There was a vast research regarding various properties of this topological index [8][9][10].
The study of determining extremal properties such as upper bounds and lower bounds of some graph invariants were subject to a large number of investigations [11][12][13][14][15].The aim of this paper is to study similar extremal properties for Zagreb eccentricity indices.In this paper we present some new upper and lower bounds of Zagreb eccentricity indices in terms of number of vertices (n), number of edges (m), radius (r), diameter (d), total eccentricity

Bounds for the First Zagreb Eccentricity Index
We now give some lower and upper bounds of first Zagreb eccentricity index.In [6]

equality if and only if G all the vertices of G are of same eccentricity.
Proof Using the Cauchy-Schwartz inequality, we get and hence using the definition of total eccentricity index and first Zagreb eccentricity index the desired result follows.Clearly in the above inequality equality holds when all the vertices of G are of same eccentricity.Theorem 2.4.Let G be a simple connected, then is the degree distance of the vertex v and is defined as Hence from the definition of first Zagreb eccentricity index, we can write , so from (2.1) we get the desired result with equality if and only if Let G be a simple connected graph with n vertices and m edges, then with equality if and only if G is a path of length one.

  
, for all Proof Again since, , with equality if and only if with equality if and only if Proof Since 0 be the number of vertices with eccentricity one in G, so the remaining vertices are of eccentricity at least two.Let be the Then from the definition of first Zagreb eccentricity index, we have from where the desired result follows.Clearly, in this theorem equality holds if and only if n n  where 0 Theorem 2.6.Let G be a simple connected graph with n vertices and m edges, then is even.
with equality if and only if all the vertices of G are of same eccentricity.Proof To prove this theorem, using the following Diaz-Metcalf inequality, we have, if a i and b i , In the above inequality equality holds if and only if or i for every .By setting i and , for Now using the definition of first Zagreb eccentricity index and total eccentricity index we get

In the above inequality equality holds if and only if all the vertices of G are of same eccentricity.
Proof Let, sum of eccentricities of the vertices adjacent to from where we get the desired result.Obviously in the above inequality equality holds if and only if all the vertices of G are of same eccentricity.Theorem 2.8.Let G be a simple connected graph on n vertices and m edges, then

In the above inequality equality holds if and only if all the vertices of G are of same eccentricity.
Proof We will prove this theorem using the following Chebyschev's inequality: Let and b are real num- bers, then n a with equality holds if and only if with equality holding if and only if we get from (2.4) from where we get the desired result.Clearly in this ine-  quality equality holds if and only if all the vertices are of same eccentricity.Theorem 2.9.Let G be a simple connected graph where all the vertices must not be of equal eccentricity, then where G consists of a number of vertices with eccentricity d and b number of vertices with eccentricity r.In the above inequality equality holds if and only if eccentricities of the vertices are equal to r + 1, r, d -1, d.Proof For any vertex with equality holds if and only if    .Now sum ve inequ lity for ming the abo a  from where we get the desired result.In the above ine- quality equality holds if and only if From the above theorem the followi esult directly follows.
, 1, Copyright © 2013 SciRes.OJDM Corollary 2.2.Let G be a simple connected graph with n vertices and m edges, then where k is the number of vertices having eccentricity equal to d or r.
the Second Zagreb Eccentricity Index

Bounds for
In [6] , so that taking natural lo oth sides and us get garithm on b ing Jensen's inequality, we which is the desired result.In this inequality equality holds if and only if all the vertices of G are of same eccentricity.em 3.4.Let G be nnected graph, the is the eccentric connectivity index of G.In the above inequality equality holds if and only if So from the definition of agreb second Z eccentricity index, we can write , the desired result follows from above.In the above inequ olds if and only if , with equality if and only if In the above inequality equality holds if and only if all the vertices of G are of same eccentricity.
Proof Since,  from where we get the desired result.Obviously i above inequality equality holds if and only if all the vertices of G are of same eccentricity.Corollary 3.2.Let G be a simple connected graph with n vertices and m edges, then 1 deg 2 Zagreb indices (M 1 (G)), the second Zagreb indices (M 2 (G)) and the eccentric eccentricity index  c G  .

3 . 5 .
Let G be a simple connected graph with   n vertices and m edges, then in [5] Xing et al. give the following result.
) we get the desired result.Since the equality (2.1) holds if and only if n so the equality holds in this result if and only if G is a path of length one which is the only complete graph as well as complete bipartite graph.
n Theorem 2.5.Let G be a simple connected graph on n vertices and 0 be the number of vertices with eccentricity one in G, then so we have and .Hence the desired result follows from above.Clearly in the above inequality equality holds if and only if all the vertices of G are of same eccentricity.
Theorem 2.7.Let G be a simple connected graph, then Das et al. give lower bounds for E 2 ( d and proved the followi from where we get the desired result.Like Corollary 2.1, Zagreb indices.It may be useful to G) and E 2 (G) indices in terms of other ol.17, 1972, pp.