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

Abstract

Classical topological indices, such as Zagreb indices ( $$M_{1}$$ and $$M_2$$ ) and the well-studied eccentric connectivity index ( $$\xi ^{c}$$ ) 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 ( $$\overline{M}_{1}$$ ) and second Zagreb coindex ( $$\overline{M}_{2}$$ ), 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 ( $$\overline{\xi }^{c}$$ ), of a connected graph. In this paper, we study the extremal problems of $$\overline{\xi }^{c}$$ 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 $$\overline{\xi }^{c}$$ in terms of several other graph parameters.