On the girth of the bipartite graph D(k,q)D(k,q)

For integer k≥2k≥2 and prime power qq, an algebraic bipartite graph D(k,q)D(k,q) of girth at least k+4k+4 was introduced by Lazebnik and Ustimenko (1995). Füredi et al. (1995) shown that the girth of D(k,q)D(k,q) is equal to k+5k+5 if kk is odd and qq is a prime power of form 1+n(k+5)/21+n(k+5)/2 and, conjectured further that D(k,q)D(k,q) has girth k+5k+5 for all odd kk and all q≥4q≥4. In this paper, we show that this conjecture is true when (k+5)/2(k+5)/2 is a power of the characteristic of FqFq.