Spectral asymmetry and odd cycles in integral graphs

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 ${\lambda }_1\ge \cdots \ge {\lambda }_n$ of $G$ satisfy ${\sum }_i({\lambda }_i+{\lambda }_{n+1-i}{)}^{+}\le 2$ and $0<{\lambda }_1+{\lambda }_n\le 1$. 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 ${\lambda }_n=1-{\lambda }_1$ and at least ${\lambda }_1-1\ge 1$ triangles, which yields the classification. We also prove that an integral graph of odd girth $g$ has at least $(g-1)!\mathrm{/}2$ cycles of length $g$, with equality for the Petersen graph when $g=5$, and we derive consequences for quantum periodicity a