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

Abstract

Let $G^σ=(G,σ)$ be a connected signed graph and $A(G^σ)$ be its adjacency matrix. The positive inertia index of $G^σ$, denoted by $p^{+}(G^σ)$, is defined as the number of positive eigenvalues of $A(G^σ)$. Assume that $G^σ$ contains at least one cycle, and let $g_{r}$ be its girth. In this paper, we prove $p^{+}(G^σ) \geq \lceil \frac {g_{r}}{2} \rceil-1$ for a signed graph $G^σ$. The extremal signed graphs corresponding to $p^{+}(G^σ) = \lceil \frac {g_{r}}{2} \rceil-1$ and $p^{+}(G^σ) =\lceil \frac {g_{r}}{2} \rceil$ 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.