Eccentric connectivity index and eccentric distance sum of some graph operations

Abstract

Let $G=(V,E)$ be a connected graph‎. ‎The eccentric connectivity index of $G$‎, ‎$xi^{c}(G)$‎, ‎is defined as‎ ‎$xi^{c}(G)=sum_{vin V(G)}deg(v)ec(v)$‎, ‎where $deg(v)$ is the‎ ‎degree of a vertex $v$ and $ec(v)$ is its eccentricity‎. ‎The‎ ‎eccentric distance sum of $G$ is defined as $xi^{d}(G)=sum_{vin‎ ‎V(G)}ec(v)D(v)$‎, ‎where $D(v)=sum_{uin V(G)}d(u,v)$‎. ‎In this paper‎, ‎we calculate the eccentric connectivity index and eccentric distance‎ ‎sum of generalized hierarchical product of graphs‎. ‎Moreover‎, ‎we‎ ‎present the exact formulae for the eccentric connectivity index of‎ ‎$F$-sum graphs in terms of some invariants of the factors‎.