  The Alon-Boppana theorem confirms that for every $\epsilon>0$ and every
integer $d\ge3$, there are only finitely many $d$-regular graphs whose second
largest eigenvalue is at most $2\sqrt{d-1}-\epsilon$. Serre gave a
strengthening showing that a positive proportion of eigenvalues of any
$d$-regular graph must be bigger than $2\sqrt{d-1}-\epsilon$. We provide a
multipartite version of this result. Our proofs are elementary and work also in
the case when graphs are not regular. In the simplest, monopartite case, our
result extends the Alon-Boppana-Serre result to non-regular graphs of minimum
degree $d$ and bounded maximum degree. The two-partite result shows that for
every $\epsilon>0$ and any positive integers $d_1,d_2,d$, every $n$-vertex
graph of maximum degree at most $d$, whose vertex set is the union of (not
necessarily disjoint) subsets $V_1,V_2$, such that every vertex in $V_i$ has at
least $d_i$ neighbors in $V_{3-i}$ for $i=1,2$, has $\Omega_\epsilon(n)$
eigenvalues that are larger than $\sqrt{d_1-1}+\sqrt{d_2-1}-\epsilon$. Finally,
we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound
$2\sqrt{d-1}-\epsilon$ can be replaced by $2\sqrt{d-1} + \delta$ for some
$\delta>0$ if graphs have bounded "global girth". On the other side of the
spectrum, if the odd girth is large, then we get an Alon-Boppana-Serre type
theorem for the negative eigenvalues as well.
