2025/05/13 by Liu, Beiyan, Duan, Fang
#FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2505.08539
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)≥ \lceil(g)/(2)\rceil-1 and extremal signed graphs corresponding to the lower bound are characterized. Furthermore, the signed graph Γ with i-(Γ)=\lceil(g)/(2)\rceil for g≥ 4 are given. As a by-product, the connected signed graphs with given positive inertia index, nullity and given girth are also determined, respectively.