Degree similar and net-degree similar signed graphs
Abstract
Two graphs are degree similar if their adjacency matrices and their degree matrices are simultaneously similar, a notion recently introduced by Godsil and Sun. We extend it to signed graphs in two ways, degree similarity and net-degree similarity, according to whether the degree or the net-degree matrix is used. We prove that signed trees are determined by their degree similarity class, but not by their net-degree similarity class. We then develop constructions of (net-)degree similar signed graphs, based on a signed version of Godsil--McKay switching, on partial transposition and on block extensions. As a byproduct, we obtain infinite families both of pairs of net-degree similar unicyclic signed graphs with non-isomorphic underlying graphs, and of unicyclic signed graphs, not necessarily regular, which are not determined by their degree similarity class. This settles the net-degree version of a problem of Godsil and Sun on unicyclic graphs.
Related
BibTeX
@misc{cavaleri2026degree,
title = {{Degree similar and net-degree similar signed graphs}},
author = {Matteo Cavaleri and Alfredo Donno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13863},
eprint = {2610.13863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13863},
}
RIS
TY - UNPB
TI - Degree similar and net-degree similar signed graphs
AU - Matteo Cavaleri
AU - Alfredo Donno
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13863
UR - https://arxiv.org/abs/2610.13863
AB - Two graphs are degree similar if their adjacency matrices and their degree matrices are simultaneously similar, a notion recently introduced by Godsil and Sun. We extend it to signed graphs in two ways, degree similarity and net-degree similarity, according to whether the degree or the net-degree matrix is used. We prove that signed trees are determined by their degree similarity class, but not by their net-degree similarity class. We then develop constructions of (net-)degree similar signed graphs, based on a signed version of Godsil--McKay switching, on partial transposition and on block extensions. As a byproduct, we obtain infinite families both of pairs of net-degree similar unicyclic signed graphs with non-isomorphic underlying graphs, and of unicyclic signed graphs, not necessarily regular, which are not determined by their degree similarity class. This settles the net-degree version of a problem of Godsil and Sun on unicyclic graphs.
ER -