Edge reconstruction of the Ihara zeta function

TL;DR

This paper demonstrates that for graphs with average degree at least 4, the Ihara zeta function can be reconstructed from edge information, revealing new spectral properties of the edge adjacency operator and related walk counts.

Contribution

It introduces the edge-reconstructibility of the Ihara zeta function for graphs with average degree ≥ 4 and analyzes spectral properties of the edge adjacency operator T.

Findings

Edge-reconstructibility of the Ihara zeta function for graphs with average degree ≥ 4

Spectral properties of the edge adjacency operator T, including symmetry and semi-simplicity

Reconstruction of non-backtracking walk counts and eigenvectors related to T

Abstract

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.