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

For integer k≥2k≥2 and prime power qq, Lazebnik and Ustimenko (1995) proposed an algebraic bipartite graph D(k,q)D(k,q) which is qq-regular, edge-transitive and of large girth. Füredi et al. (1995) conjectured that D(k,q)D(k,q) has girth k+5k+5 for all odd kk and all q≥4q≥4 and, shown that this conjecture is true for the case that (k+5)/2(k+5)/2 divides q−1q−1. Cheng et al. (2014) shown that this conjecture is true for the case that (k+5)/2(k+5)/2 is an arbitrary power of the characteristic of FqFq. In this paper, we propose a generalization for the binomial coefficients and show that this conjecture is true when (k+5)/2(k+5)/2 is the product of an arbitrary factor of q−1q−1 and an arbitrary power of the characteristic of FqFq.