  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.
