Characterizing the positive inertia index of connected signed graphs in terms of girth

Abstract

Let $G^{\sigma }=(G,\sigma )$ be a connected signed graph and $A(G^{\sigma })$ be its adjacency matrix. The positive inertia index of $G^{\sigma }$, denoted by $p^{+}(G^{\sigma })$, is defined as the number of positive eigenvalues of $A(G^{\sigma })$. Assume that $G^{\sigma }$ contains at least one cycle, and let $g_r$ be its girth. In this paper, we prove $p^{+}(G^{\sigma })\ge \lceil \frac{g_r}{2}\rceil -1$ for a signed graph $G^{\sigma }$. The extremal signed graphs corresponding to $p^{+}(G^{\sigma })=\lceil \frac{g_r}{2}\rceil -1$ and $p^{+}(G^{\sigma })=\lceil \frac{g_r}{2}\rceil$ are characterized, respectively. The results presented in this article extend the recent work on ordinary graphs by Duan and Yang (Linear Algebra Appl., 2024) to the context of signed graphs.

10 pages, 2 figures