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Zagreb eccentricity indices of the generalized hierarchical product graphs and their applications. (English) Zbl 1406.05021

Summary: Let \(G\) be a connected graph. The first and second Zagreb eccentricity indices of \(G\) are defined as \(M_1^\ast(G)=\sum_{v\in V(G)}\varepsilon^2_G(v)\) and \(M_2^\ast(G)=\sum_{uv\in E(G)}\varepsilon_G(u)\varepsilon_G(v)\), where \(\varepsilon_G(v)\) is the eccentricity of the vertex \(v\) in \(G\) and \(\varepsilon^2_G(v)=(\varepsilon_G(v))^2\). Suppose that \(G(U)\sqcap H(\emptyset\neq U\subseteq V(G))\) is the generalized hierarchical product of two connected graphs \(G\) and \(H\). In this paper, the Zagreb eccentricity indices \(M_1^\ast\) and \(M_2^\ast\) of \(G(U)\sqcap H\) are computed. Moreover, we present explicit formulas for the \(M_1^\ast\) and \(M_2^\ast\) of \(S\)-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.

MSC:

05C07 Vertex degrees
05C12 Distance in graphs
05C76 Graph operations (line graphs, products, etc.)
05C90 Applications of graph theory
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