On the characterization of some algebraically defined bipartite graphs of girth eight

For any field F and polynomials f(2), f(3) is an element of F[x, y], let G(F)(f(2), f(3)) denote the bipartite graph with vertex partition P boolean OR L, where P and L are two copies of F-3, and (p(1), p(2), p(3)) is an element of P is adjacent to [l(1), l(2), l(3)]. L if and only if p(2)+l(2) = f(2)(p(1), l(1)) and p(3)+l(3) = f(3)(p(1), l(1)). The graph Gamma(3)(F) = Gamma(F)(xy, xy(2)) is known to be of girth eight. When F = F-q is a finite field of odd characteristic or F = F-infinity is an algebraically closed field of characteristic zero, the graph Gamma(3)(F) is conjectured to be the unique one with girth at least eight among those Gamma(F)(f(2), f(3)) up to isomorphism. This conjecture has been confirmed for the case that both f(2), f(3) are monomials over Fq, and for the case that at least one of f(2), f(3) is a monomial over F-infinity. If one of f(2), f(3). F-q[x, y] is a monomial, it has also been proved the existence of a positive integer M such that G = Gamma(FqM) (f(2), f(3)) is isomorphic to Gamma(3)(F-qM) provided G has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials f(2), f(3) is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials f(2), f(3) are characterized completely. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study investigates the properties of bipartite graphs based on specific polynomial structures and discusses and proves the isomorphism in specific cases, demonstrating the validity of the corresponding conclusions.