  A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the
vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this
paper, we give an improved new Moore bound for an infinite family of such
graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol,
and F\`abrega.\\ Besides, we propose some constructions of bipartite biregular
graphs with diameter $d$ and large number of vertices $N(r_1,r_2;d)$, together
with their spectra. In some cases of diameters $d=3$, $4$, and $5$, the new
graphs attaining the Moore bound are unique up to isomorphism.
