New results and open problems on subgraph centrality

TL;DR

This paper reviews recent advances and open problems related to subgraph centrality, a spectral measure used to analyze node importance in various complex networks across multiple disciplines.

Contribution

It presents new mathematical results and highlights unresolved questions concerning subgraph centrality and related walk-based centrality measures.

Findings

New theoretical results on subgraph centrality

Identification of open problems in spectral graph theory

Connections to number theory and combinatorics

Abstract

Subgraph centrality, introduced by Estrada and Rodr\'iguez-Vel\'azquez in [12], has become a widely used centrality measure in the analysis of networks, with applications in biology, neuroscience, economics and many other fields. It is also worthy of study from a strictly mathematical point of view, in view of its connections to topics in spectral graph theory, number theory, analytic matrix functions, and combinatorics. In this paper we present some new results and a list of open questions about subgraph centrality and other node centrality measures based on graph walks.