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# Eccentric connectivity index and eccentric distance sum of some graph operations

Author(s): Buzohragul Eskender | Elkin Vumar

Journal: Transactions on Combinatorics
ISSN 2251-8657

Volume: 2;
Issue: 1;
Start page: 103;
Date: 2013;
Original page

Keywords: Eccentric connectivity index | eccentric distance sum | generalized hierarchical product | \$F\$-sum graphs

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_{G}(u,v)\$ and \$d_{G}(u,v)\$ is the distance between \$u\$ and \$v\$ in \$G\$. In this paper, we calculate the eccentric connectivity index and eccentric distance sum of generalized hierarchical product of graphs. Moreover, we present explicit formulae for the eccentric connectivity index of \$F\$-sum graphs in terms of some invariants of the factors. As applications, we present exact formulae for the values of the eccentric connectivity index of some graphs of chemical interest such as \$C_{4}\$ nanotubes, \$C_{4}\$ nanotoris and hexagonal chains.