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On the spectral closeness and residual spectral closeness of graphs. (English) Zbl 1497.05169

Summary: The spectral closeness of a graph \(G\) is defined as the spectral radius of the closeness matrix of \(G\), whose \((u, v)\)-entry for vertex \(u\) and vertex \(v\) is \(2^{-d_G (u,v)}\) if \(u \neq v\) and 0 otherwise, where \(d_G (u, v)\) is the distance between \(u\) and \(v\) in \(G\). The residual spectral closeness of a nontrivial graph \(G\) is defined as the minimum spectral closeness of the subgraphs of \(G\) with one vertex deleted. We propose local grafting operations that decrease or increase the spectral closeness and determine those graphs that uniquely minimize and/or maximize the spectral closeness in some families of graphs. We also discuss extremal properties of the residual spectral closeness.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C35 Extremal problems in graph theory
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
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