Spectral asymmetry and odd cycles in integral graphs
Abstract
We prove that the triangle C3 is the only integral unicyclic graph with an odd cycle. The main tool is a spectral inequality that does not require integrality: if G is a connected non-bipartite graph and G−e is bipartite for some edge e, then the eigenvalues λ1≥⋯≥λn of G satisfy ∑i(λi+λn+1−i)+≤ 2 and 0<λ1+λn≤ 1. It follows from comparing A(G) with the signed adjacency matrix obtained by reversing the sign of e, which is similar to −A(G), through a common rank-one perturbation and Weyl's monotonicity theorem. For integral graphs in this class, an arithmetic lemma then gives λn=1−λ1 and at least λ1−1≥1 triangles, which yields the classification. We also prove that an integral graph of odd girth g has at least (g−1)!/2 cycles of length g, with equality for the Petersen graph when g=5, and we derive consequences for quantum periodicity a
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BibTeX
@misc{braga2026spectral,
title = {{Spectral asymmetry and odd cycles in integral graphs}},
author = {Rodrigo O. Braga and Jean Carlo Moraes and Matheus C. Santos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22773},
eprint = {2610.22773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22773},
}
RIS
TY - UNPB
TI - Spectral asymmetry and odd cycles in integral graphs
AU - Rodrigo O. Braga
AU - Jean Carlo Moraes
AU - Matheus C. Santos
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22773
UR - https://arxiv.org/abs/2610.22773
AB - We prove that the triangle C_3 is the only integral unicyclic graph with an odd cycle. The main tool is a spectral inequality that does not require integrality: if G is a connected non-bipartite graph and G−e is bipartite for some edge e, then the eigenvalues λ_1≥⋯≥λ_n of G satisfy ∑_i(λ_i+λ_n+1−i)^+≤ 2 and 0<λ_1+λ_n≤ 1. It follows from comparing A(G) with the signed adjacency matrix obtained by reversing the sign of e, which is similar to −A(G), through a common rank-one perturbation and Weyl's monotonicity theorem. For integral graphs in this class, an arithmetic lemma then gives λ_n=1−λ_1 and at least λ_1−1≥1 triangles, which yields the classification. We also prove that an integral graph of odd girth g has at least (g−1)!/2 cycles of length g, with equality for the Petersen graph when g=5, and we derive consequences for quantum periodicity a
ER -