On Zagreb Eccentricity Indices

Zagreb eccentricity indices were proposed analogously to Zagreb indices already known and used for almost forty years. For a connected graph, the first Zagreb eccentricity index is defined as the sum of the squares of the eccentricities of the vertices, and the second Zagreb eccentricity index is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. We report mathematical properties, especially lower and upper bounds of trees and general graphs in terms of graph invariants and the corresponding extremal graphs, Nordhaus-Gaddum-type results, and the ordering of trees with small and large Zagreb eccentricity indices. (doi: 10.5562/cca1801)


INTRODUCTION
Let G be a connected graph with vertex set V(G) and edge set E(G).For a vertex ( ), u V G  ( ) G e u or u e denotes the eccentricity of u in G, which is the length of a path from u to a vertex v that is farthest from u, i.e., max{ ( , ) : where ( , | ) d u v G denotes the distance between u and v in G. 1 The Zagreb eccentricity indices are introduced in an analogy with the Zagreb indices [2][3][4] by replacing the vertex degrees with the vertex eccentricities.Thus, the first Zagreb eccentricity index of a graph G is defined as: and the second Zagreb eccentricity index of G is defined as: Note that degrees are "local properties", while eccentricities are "global properties" of the vertices.These two indices are recently introduced by Vukičević and Graovac. 5However, it should be pointed out that there were a number of eccentricity-based molecular descriptors already proposed in the literature, 6,7 such as the eccentric connectivity index 8,9 (see References 10-12 for recent results) and the eccentric adjacency index. 13Vukičević and Graovac have shown in their paper 5 if G is a tree or a unicyclic graph but is not true for bicyclic graphs.
In the present paper, we give lower and upper bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed diameter, and lower bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed matching number, and characterize the extremal cases, and determine the n-vertex trees with respectively the minimum, secondminimum and third-minimum, as well as the maximum, second-maximum and third-maximum indices 1 ξ and 2 ξ for 6. n  We also give lower and upper bounds for the first and the second Zagreb eccentricity indices of connected graphs in terms of graph invariants such as the number of vertices, the number of edges, the radius and the diameter, and give the Nordhaus-Gaddum-type results. 14

PRELIMINARIES
For a connected graph G, the radius r(G) and the diameter D(G) are, respectively, the minimum and maximum eccentricities among the vertices of G. 1 For a graph G and a subset E' of its edge set (E* of the edge set of its complement, respectively), G -E' (G + E*, respectively) denotes the graph formed from G by deleting (adding, respectively) edges from E' (E* respectively).For a graph G with ( ), u V G  G -u denotes the graph formed from G by deleting the vertex u (and its incident edges).

RESULTS FOR TREES
In this section, we give lower and upper bounds for the first and the second Zagreb eccentricity indices of n-vertex trees with fixed diameter, and lower bounds for the first and the second Zagreb eccentricity indices of nvertex trees with fixed matching number.We also determine the n-vertex trees with, respectively, the minimum, second-minimum and third-minimum, as well as the maximum, second-maximum and third-maximum indices 1 ξ and 2 ξ for 6. n  Lemma 1.Let u be a vertex of a tree Q with at least two vertices.For integer 1, a  let G 1 be the tree obtained from Q by attaching a star S a+1 at its center v to , u G 2 the tree obtained from Q by attaching a + 1 pendant vertices to , u see Figure 1.pendant vertices, respectively, to the two end vertices of the path The matching number of G is the number of edges of a maximum matching in .
is incident with an edge of M , then v is said to be Msaturated, and if every vertex of G is M-saturated, then M is a perfect matching.T T  Now we use the result for the first and the second Zagreb eccentricity indices of the n-vertex trees with fixed diameter to determine the n-vertex trees with small and large indices 1 ξ and 2 .ξ Proposition 5.Among the n-vertex trees, n S for 3, n  the trees in ,3 L n for 4, n  and the trees in ,4 L n for 5 n  are, respectively, the unique trees with the minimum, second-minimum and third-minimum indices 1 ξ and 2 , ξ the first Zagreb eccentricity indices of which are equal to 4  let n P be the tree formed by attaching a pendant vertex to the neighbor of an end-vertex of the path 1 .
n P  For 6, n  let n P be the tree formed by attaching a pendant vertex to the second neighbor of an end-vertex of the path

RESULTS FOR GENERAL CONNECTED GRAPHS
In this section, we give lower and upper bounds for the first and the second Zagreb eccentricity indices of connected graphs in terms of graph invariants such as the number of vertices, the number of edges, the radius and the diameter, and we also come up with the Nordhaus-Gaddum-type results. 14Moreover, among the n-vertex connected graphs, we establish lower and upper bounds for the first Zagreb eccentricity index, and lower bound for the second Zagreb eccentricity index, respectively, and characterize the extremal cases.

Proposition 7.
Let G be a connected graph with n vertices and m edges.Then: with any equality if and only if G is a self-centered graph.
Proposition 8. Let G be a connected graph with 2 n  vertices.Then:

Figure 1 .
Figure 1.The trees G 1 and G 2 in Lemma 1.Figure 2. The structure of trees in L n,d with odd d.

Figure 2 .
Figure 1.The trees G 1 and G 2 in Lemma 1.Figure 2. The structure of trees in L n,d with odd d.

Figure 3 .
Figure 3.The tree in L n,d with even d.

Figure 6 .
Figure 6.The graph B n .