Some classes of connected signed graphs with girth $g$ and negative inertia index $\lceil\frac{g}{2}\rceil+1$

Abstract

Let $Γ$ be a signed graph. The number of negative eigenvalues of the adjacency matrix of $Γ$ is called the negative inertia index of $Γ$, which is denoted by $i_-(Γ)$. The length of the shortest cycle contained in $Γ$ is called the girth of $Γ$, and it is denoted by $g$. In this paper, we give some classes of connected signed graphs $Γ$ which satisfy the condition $i_-(Γ)=\lceil\frac{g}{2}\rceil+1$.

Figures (5)

• Figure 1: The signed graphs $(B(4,3,4),+)$, $\Gamma'$, $H_4^{\sigma_2}$, $H_5^{\sigma}$, $(B(4,3,5),\sigma_2)$, $(B(5,2,5),+)$, $(B(5,3,5),\sigma)$, $(B(5,4,5),\sigma)$ and $(B(5,5,5),+)$
• Figure 2: The signed graphs $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$
• Figure 3: The signed graphs $H_1^\sigma$ and $H_2^\sigma$
• Figure 4: The signed graphs $\Gamma_4$, $\Gamma_5$, $(B(5,3,5),+)$, $(B(5,3,5),-)$, $(B(4,4,5),-)$, $\Gamma_6$, $\Gamma_7$, $(B(5,4,5),+)$, $(B(5,4,5),-)$, $\Gamma_4$, $\Gamma_8$, $\Gamma_9$, $(B(5,4,6),+)$, $(B(6,3,6),{\sigma_1})$, $\Gamma_{10}$ and $\Gamma_{11}$
• Figure 5: The signed graphs $H_3^\sigma$, $(B(4,4,5),+)$, $(B(5,3,6),\sigma)$, $(B(6,2,6),\sigma)$, $(B(4,4,6),\sigma)$, $(B(5,4,6),\sigma)$, $(B(6,3,6),\sigma)$ and $(B(6,6,6),\sigma)$

Theorems & Definitions (23)

• Lemma 2.1: R.A.Horn
• Lemma 2.2
• Lemma 2.3: L.D
• Lemma 2.4: G.H.Yu
• Lemma 2.5
• Lemma 2.6: GuiHai.Yu, G.H.Yu, G.H.Yu1
• Lemma 2.7
• proof
• Theorem 2.1: Yaoping.Hou
• Lemma 2.8: FanY.Z