The relationship between the negative inertia index of graph $G$ and its girth $g$ and diameter $d$

Abstract

Let $G$ be a simple connected graph. We use $n(G)$, $p(G)$, and $η(G)$ to denote the number of negative eigenvalues, positive eigenvalues, and zero eigenvalues of the adjacency matrix $A(G)$ of $G$, respectively. In this paper, we prove that $2n(G)\geq d(G) + 1$ when $d(G)$ is odd, and $n(G) \geq \lceil \frac{g}{2}\rceil - 1$ for a graph containing cycles, where $d(G)$ and $g$ are the diameter and girth of the graph $G$, respectively. Furthermore, we characterize the extremal graphs for the cases of $2n(G) = d(G) + 1$, $n(G) = \lceil \frac{g}{2}\rceil$, and $n(G) = \lceil \frac{g}{2}\rceil - 1$.

Figures (6)

• Figure 1: $G_0$ and $G_1$
• Figure 2: $H_i$ for $5\leq i\leq14$
• Figure 3: $G_2$ and $G_3$
• Figure 4: $G[V(C)\cup\{x,x_1,x_2,y_1,y_2\}]$
• Figure 5: $G_8, B(5,2,5)$ and $B(4,3,5)$

Theorems & Definitions (29)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• proof
• Theorem 2.9