We show that if a graph $G$ has average degree $\overline d \geq 4$, 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 $\overline d>4$, 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.
@article{10_37236_5909,
author = {Gunther Cornelissen and Janne Kool},
title = {Edge reconstruction of the {Ihara} zeta function},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/5909},
zbl = {1391.05162},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5909/}
}
TY - JOUR
AU - Gunther Cornelissen
AU - Janne Kool
TI - Edge reconstruction of the Ihara zeta function
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/5909/
DO - 10.37236/5909
ID - 10_37236_5909
ER -
%0 Journal Article
%A Gunther Cornelissen
%A Janne Kool
%T Edge reconstruction of the Ihara zeta function
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/5909/
%R 10.37236/5909
%F 10_37236_5909
Gunther Cornelissen; Janne Kool. Edge reconstruction of the Ihara zeta function. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/5909
