  Counting short cycles in bipartite graphs is a fundamental problem of
interest in the analysis and design of low-density parity-check (LDPC) codes.
The vast majority of research in this area is focused on algorithmic
techniques. Most recently, Blake and Lin proposed a computational technique to
count the number of cycles of length $g$ in a bi-regular bipartite graph, where
$g$ is the girth of the graph. The information required for the computation is
the node degree and the multiplicity of the nodes on both sides of the
partition, as well as the eigenvalues of the adjacency matrix of the graph
(graph spectrum). In this paper, the result of Blake and Lin is extended to
compute the number of cycles of length $g+2, \ldots, 2g-2$, for bi-regular
bipartite graphs, as well as the number of $4$-cycles and $6$-cycles in
irregular and half-regular bipartite graphs, with $g \geq 4$ and $g \geq 6$,
respectively.
