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

TL;DR

This paper proves a conjecture relating vertex importance measures in networks, showing that vertices with equal subgraph centrality are cospectral and share degree and eigenvector centralities, advancing understanding of network vertex distinctions.

Contribution

It proves Estrada's conjecture that equal subgraph centrality implies cospectrality and related properties, generalizing previous results in network theory.

Findings

Vertices with equal $eta$-subgraph centrality are necessarily cospectral.

Such vertices have the same degree and eigenvector centralities.

The results settle a conjecture of Estrada and its generalization.

Abstract

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.