Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the \beta-subgraph centrality, which is based on the exponential of the matrix \beta A, where A is the adjacency matrix of the graph and \beta is a real parameter ("inverse temperature"). We prove that for algebraic \beta, two vertices with equal \beta-subgraph centrality have necessarily the same number of closed walks of agiven length passing through them . We further show that two such vertices must have the same degree and eigenvector centralities. Our results settle a conjecture of Estrada and a generalization of it due to Kloster, Kr\'al and Sullivan. We also discuss possible extensions of our results.
Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the \beta-subgraph centrality, which is based on the exponential of the matrix \beta A, where A is the adjacency matrix of the graph and \beta is a real parameter ("inverse temperature"). We prove that for algebraic \beta, two vertices with equal \beta-subgraph centrality are necessarily cospectral . We further show that two such vertices must have the same degree and eigenvector centralities. Our results settle a conjecture of Estrada and a generalization of it due to Kloster, Kr\'al and Sullivan. We also discuss possible extensions of our results.
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have necessarily the same number of closed walks of agiven length passing through them
are necessarily cospectral