  For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph
$D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in
extremal graph theory but also in coding theory and cryptography. This graph is
$q$-regular, edge-transitive and of girth at least $k+4$.
  Its exact girth $g=g(D(k,q))$ was conjectured in 1995 to be $k+5$ for odd $k$
and $q\geq4$.
  This conjecture was shown to be valid in 2016 when $\frac{k+5}{2}|_p(q-1)$,
where $p$ is the characteristic of $\mathbb{F}_q$ and $m|_pn$ means that $m$
divides $p^r n$ for some nonnegative integer $r$. In this paper, for $t\geq 1$
we prove that (a) $g(D(4t+2,q))=g(D(4t+1,q))$; (b) $g(D(4t+3,q))=4t+8$ if
$g(D(2t,q))=2t+4$; (c) $g(D(8t,q))=8t+4$ if $g(D(4t-2,q))=4t+2$; (d)
$g(D(2^{s+2}(2t-1)-5,q))=2^{s+2}(2t-1)$ if $p\geq 3$, $(2t-1)|_p(q-1)$ and
$2^s\|(q-1)$.
  A simple upper bound for the girth of $D(k,q)$ is proposed in the end of this
paper.
