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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.

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