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arXiv · 2505.08539

Connected signed graphs with given inertia indices and given girth

Beiyan Liu·Fang Duan
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Abstract

Suppose that $Γ=(G, σ)$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $Γ$ are called positive inertia index, negative inertia index and nullity of $Γ$, which are denoted by $i_+(Γ)$, $i_-(Γ)$ and $η(Γ)$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $Γ$. We study relationships between the girth and the negative inertia index of $Γ$ in this article. We prove $i_{-}(Γ)\geq \lceil\frac{g}{2}\rceil-1$ and extremal signed graphs corresponding to the lower bound are characterized. Furthermore, the signed graph $Γ$ with $i_{-}(Γ)=\lceil\frac{g}{2}\rceil$ for $g\geq 4$ are given. As a by-product, the connected signed graphs with given positive inertia index, nullity and given girth are also determined, respectively.

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Beiyan Liu, Fang Duan. 2025-05-13. Connected signed graphs with given inertia indices and given girth. https://arxiv.org/abs/2505.08539

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