Connected signed graphs with given inertia indices and given girth

TL;DR

Problem: relate girth $g$ to inertia indices (i_minus, i_plus, and eta) of connected signed graphs. Approach: apply Sylvester's law of inertia, interlacing, switching, and constructive operations (twin vertices, pendant stars) on canonical unicyclic cores to derive bounds and equalities. Contributions: complete classifications of graphs with given girth and inertia profiles, including cycles with sign constraints, canonical unicyclic structures, and explicit nullity bounds; extended results cover all inertia configurations. Impact: provides a unified spectral-structural taxonomy for signed graphs with prescribed girth, enabling sharper spectral predictions and potential applications in chemistry-inspired network models.

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.

Figures (5)

• Figure 1: The signed graphs $G_1^\sigma$ and $G_2^\sigma$
• Figure 2: The canonical signed unicyclic graphs $K_1^\sigma$ and $K_2^\sigma$
• Figure 3: The signed graphs $(B(4,3,4),\sigma)$, $(B(4,4,4),\sigma_1)$, $(B(4,4,4),\sigma_2)$, $H_4^\sigma$, $(B(4,3,5),$$\sigma)$ and $(B(4,4,5),\sigma)$
• Figure 4: The signed graphs $H_1^\sigma$, $H_2^\sigma$, $(H_3,+)$, $H_4^{\sigma_1}$, $(B(4,3,5),\sigma_1)$, $(B(4,4,5),\sigma_1)$ and $(B(4,4,5),\sigma_2)$.
• Figure 5: The signed graphs $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $(B(4,3,4),+)$, $\Gamma_4$, $H_4^{\sigma_2}$, $H_5^\sigma$, $(B(4,3,5),\sigma_2)$, $(B(5,2,5),$$+)$, $(B(5,5,5),+)$, $(B(5,3,5),\sigma)$ and $(B(5,4,5),\sigma)$.

Theorems & Definitions (33)

• Theorem 1.1: Yaoping.Hou
• Lemma 1.1: FanY.Z
• Lemma 2.1
• Lemma 2.2: R.A.Horn
• Theorem 2.1: D.Cvetkovi
• Lemma 2.3
• Lemma 2.4: GuiHai.Yu, G.H.Yu, G.H.Yu1
• Lemma 2.5: G.H.Yu
• Lemma 2.6: Fang.D
• Theorem 2.2: Fang.D