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

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