  Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$.
The vertex connectivity of $G$, written $\kappa(G)$, is the minimum size of a
vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G)
\le \kappa(G)$ for a non-complete simple graph $G$; for this reason $\mu_2(G)$
is called the "algebraic connectivity" of $G$. We extend his result to
multigraphs. For a pair of vertices $u$ and $v$, let $m(u,v)$ be the number of
edges with endpoints $u$ and $v$. For a vertex $v$, let $m(v)=\max_{u \in N(v)}
m(v,u)$, where $N(v)$ is the set of neighbors of $v$, and let $m(G)=\max_{v \in
V(G)} m(v)$. We prove that for any multigraph $G$ whose underlying graph is not
a complete graph, $\mu_2(G) \le \kappa(G) m(G)$.
  We also prove that for any $d$-regular multigraph $G$ whose underlying graph
is not the complete graph with 2 vertices, if $\mu_2(G) > \frac d4$, then $G$
is 2-connected. For $t\ge2$ and infinitely many $d$, we construct $d$-regular
multigraphs $H$ with $\mu_2(H)=d$, $\kappa(H)=t$, and $m(H)=\frac dt$. These
graphs show that the inequality $\mu_2(G) \le \kappa(G) m(G)$ is sharp. In
addition, we prove that if $G$ is a $d$-regular multigraph whose underlying
graph is not a complete graph, then $\mu_2(G) \le d$; equality holds for the
graphs in the construction.
