On the uniqueness of some girth eight algebraically defined graphs, Part II

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

“…The primary goal of study in [5] was to ascertain that when R is a finitehttps://www.overleaf.com/project/5db074df0a3a250001 field F q of odd order q, the only (up to isomorphism) girth eight graph Γ R (f, g), where f and g are monomials in R[X, Y ], is Γ R (XY, XY 2 ). 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 ).…”

“…Of particular interest is the question whether all such graphs are isomorphic to Γ R (XY, XY 2 ); this stands in contrast to the situation R = F q with q odd (as in [5]) and R = C (as in [11,12]), where it is known that all monomial graphs of girth eight are isomorphic to Γ R (XY, XY 2 ). We do not for instance know whether Γ R (XY, XY 2 ) is isomorphic to Γ R (XY, XY 4 ).…”

“…For a detailed explanation of this construction, see [11] (concluding remarks), [13] (Sections 4.2 and 4.3), and references therein. This study was later generalized to algebraically defined graphs over fields of characteristic zero, including the field of complex numbers (as in [11,12]) and the field of real numbers (as in this paper).…”

“…Since ( 1) and ( 2) appear repeatedly throughout this paper, we will introduce the following notation used, e.g., in [2,3,4,11,12]:…”

“…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.…”