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

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