Characterizing the positive inertia index of connected signed graphs in terms of girth
Suliman Khan, Sakander Hayat, Mohammed J. F. Alenazi
TL;DR
This work bounds the positive inertia index $p^{+}(G^{\sigma})$ of a connected signed graph by its girth $g_r$ and derives sharp extremal classifications. By extending interlacing techniques and cycle-signature analysis from ordinary graphs to signed graphs, the authors prove $p^{+}(G^{\sigma}) \geq \left\lceil \frac{g_r}{2} \right\rceil - 1$ and characterize when equality occurs, revealing two primary extremal families: balanced cycles with specific parity and balanced complete multipartite graphs. The results further sharpen to classify all graphs with $p^{+}(G^{\sigma}) = \left\lceil \frac{g_r}{2} \right\rceil$ and provide a detailed taxonomy of canonical unicyclic and certain bicyclic signed graphs achieving this bound. Overall, the paper extends Duan and Yang's 2024 findings for ordinary graphs to the signed-graph setting, enriching the relationship between inertia indices and girth in spectral graph theory.
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.