New distance-based graph invariants and relations among them

The eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs. In this paper, we give some results on average (edge, vertex-edge) eccentricities of graph invariants. There are many examples of graph parameters, specially based on distances which are applicable in chemistry. The Wiener index is probably the most studied graph invariant in both theoretical and practical meanings (cf. [4,6–11,22]). Apart from the Wiener index, we will consider some other related indices. One of them is the average eccentricity defined as the average of eccentiricities of all vertices in a graph. This was first introduced by Buckley and Harary [1], and then not many works have been done except undirect studies [3,5]. Let us consider a communication network modelled by a graph with vertices representing the codes of the network and edges representing the links between them. One might want to minimize the average, taken over all the nodes in the system, of the maximum time delay of a message emanating from it. This is the average eccentricity of the corresponding graph. Throughout this paper, it will be only considered simple connected graphs. We remind that, for a graph G, the notation V ¼ VðGÞ and E ¼ EðGÞ will denote the vertex and edge sets, respectively. Also, while the order of G is defined by n ¼ jVðGÞj, the size of G is defined by m ¼ jEðGÞj. The following material will be needed in this paper as the preliminary material. For the set of non-negative integers Z þ , let us consider the mappings