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Centrality measures are used in network science to identify the mostimportant vertices for transmission of information and dynamics on a graph. Oneof 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 realparameter ("inverse temperature"). We prove that for algebraic $\beta$, twovertices with equal $\beta$-subgraph centrality are necessarily cospectral. Wefurther show that two such vertices must have the same degree and eigenvectorcentralities. Our results settle a conjecture of Estrada and a generalizationof it due to Kloster, Kr\'al and Sullivan. We also discuss possible extensionsof our results.

Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations