Girth of the algebraic bipartite graph D(k,q)

Abstract

For integer k≥2 and prime power q, the algebraic bipartite graph D(k,q) proposed by Lazebnik and Ustimenko in 1993 is useful 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≥4. This conjecture was shown to be valid in 2016 when k+52|p(q−1), where p is the characteristic of Fq and m|pn means that m divides prn for some nonnegative integer r. In this paper, for t≥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(2s+2t−5,q))=2s+2t if p≥3, 2s|(q−1), 2s+1∤(q−1), 2∤t and t|p(q−1). A simple upper bound for the girth of D(k,q) is proposed in the end of this paper.