The Total Eccentricity Sum of Non-adjacent Vertex Pairs in Graphs.

Classical topological indices, such as Zagreb indices ( M 1 and M 2 ) and the well-studied eccentric connectivity index ( ξ c ) directly or indirectly consider the total contribution of all edges in a graph. By considering the total degree sum of all non-adjacent vertex pairs in a graph, Ashrafi et al. (Discrete Appl Math 158:1571–1578, 2010) proposed two new Zagreb-type indices, namely the first Zagreb coindex ( M ¯ 1 ) and second Zagreb coindex ( M ¯ 2 ), respectively. Motivated by Ashrafi et al., we consider the total eccentricity sum of all non-adjacent vertex pairs, which we call the eccentric connectivity coindex ( ξ ¯ c ), of a connected graph. In this paper, we study the extremal problems of ξ ¯ c for connected graphs of given order, connected graphs of given order and size, and the trees, unicyclic graphs, bipartite graphs containing cycles and triangle-free graphs of given order, respectively. Additionally, we establish various lower bounds for ξ ¯ c in terms of several other graph parameters. [ABSTRACT FROM AUTHOR]