The eccentric connectivity polynomial of some graph operations. (English) Zbl 1284.05153

Ashrafi, A. R.; Ghorbani, M.; Hossein-Zadeh, M. A.

Serdica J. Comput. 5, No. 2, 101-116 (2011).

Summary: The eccentric connectivity index of a graph \(G\), \(\xi^C\), was proposed by V. Sharma, R. Goswami and A. K. Madan [J. Chem. Inf. Comput. Sci. 37, 273–282 (1997)]. It is defined as \[ \xi^C(G)= \sum_{u\in V(G)} \deg_G(u)\varepsilon_G(u), \] where \(\deg_G(u)\) denotes the degree of the vertex in \(G\) and \[ \varepsilon_G(u)= \text{Max}\{d(u,x)\mid x\in V(G)\}. \] The eccentric connectivity polynomial is a polynomial version of this topological index.
In this paper, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented.

MSC:

05C40	Connectivity
05C76	Graph operations (line graphs, products, etc.)

Keywords:

graph operator; topological index; eccentric connectivity polynomial