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

Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

“…A similar assertion was proven in [11] and [12]: whenever R is an algebraically closed field of characteristic zero, the only (up to isomorphism) graph Γ R (X k Y m , g) of girth at least eight, where k, m ∈ N and g ∈ R[X, Y ], is Γ R (XY, XY 2 ). It was proven [15] that given any polynomials f ∈ F q [X], g ∈ F q [Y ], and h ∈ F q [X, Y ], there exists a positive integer M depending on the degrees of f , g, and h, such that any graph Γ R (f g, h) with R = F q M of girth at least eight is isomorphic to Γ R (XY, XY 2 ); it was also proven that when R is any algebraically closed field of characteristic zero, the only graph Γ R (f (X)g(Y ), h(X, Y )) (up to isomorphism) of girth at least eight is Γ R (XY, XY 2 ). Finally, there are no graphs of girth eight or more in the two-dimensional real case; see [4].…”

“…This work was expanded upon by Kronenthal [10], and the conjecture was ultimately proven by Hou, Lappano, and Lazebnik [5]. In addition, Kronenthal and Lazebnik [11] and Kronenthal, Lazebnik, and Williford [12] studied families of polynomial graphs over algebraically closed fields of characteristic zero and applied some of their techniques to graphs over finite fields; these results were recently extended by Xu, Cheng, and Tang [15]. Moreover, Ganger, Golden, Kronenthal, and Lyons [4] studied a two-dimensional analogue over the real numbers.…”