Connectivity and eigenvalues of graphs with given girth or clique number

Abstract

Let $\kappa'(G)$, $\kappa(G)$, $\mu_{n-1}(G)$ and $\mu_1(G)$ denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of $G$, respectively. In this paper, we prove that for integers $k\geq 2$ and $r\geq 2$, and any simple graph $G$ of order $n$ with minimum degree $\delta\geq k$, girth $g\geq 3$ and clique number $\omega(G)\leq r$, the edge-connectivity $\kappa'(G)\geq k$ if $\mu_{n-1}(G) \geq \frac{(k-1)n}{N(\delta,g)(n-N(\delta,g))}$ or if $\mu_{n-1}(G) \geq \frac{(k-1)n}{\varphi(\delta,r)(n-\varphi(\delta,r))}$, where $N(\delta,g)$ is the Moore bound on the smallest possible number of vertices such that there exists a $\delta$-regular simple graph with girth $g$, and $\varphi(\delta,r) = \max\{\delta+1,\lfloor\frac{r\delta}{r-1}\rfloor\}$. Analogue results involving $\mu_{n-1}(G)$ and $\frac{\mu_1(G)}{\mu_{n-1}(G)}$ to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [Linear Algebra Appl. 439 (2013) 3777--3784], [Linear Algebra Appl. 578 (2019) 411--424], [Linear Algebra Appl. 579 (2019) 72--88], [Appl. Math. Comput. 344-345 (2019) 141--149] and [Electronic J. Linear Algebra 34 (2018) 428--443] are improved or extended.