  Recently, working on the Tanner graph which represents a low density parity
check (LDPC) code becomes an interesting research subject. Finding the number
of short cycles of Tanner graphs motivated Blake and Lin to investigate the
multiplicity of cycles of length girth in bi-regular bipartite graphs, by using
the spectrum and degree distribution of the graph. Although there were many
algorithms to find the number of cycles, they preferred to investigate in a
computational way. Dehghan and Banihashemi counted the number of cycles of
length $g+2$ and $g+4,$ where $G$ is a bi-regular bipartite graph and $g$ is
the length of the girth $G.$ But they just proposed a descriptive technique to
compute the multiplicity of cycles of length less than $2g$ for bi-regular
bipartite graphs. In this paper, we find the number of cycles of length less
than $2g$ by using spectrum and degree distribution of a bi-regular bipartite
graph such that the formula depends only on the partitions of positive integers
and the number of closed cycle-free walks from a vertex of the bi-regular
bipartite graph.
