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.


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