On the uniqueness of some girth eight algebraically defined graphs

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

“…This motivated exploration of the q odd case, as the existence of non-isomorphic girth eight algebraically defined graphs could lead to the construction of new generalized quadrangles. 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).…”

“…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]:…”

“…We end this section with the following isomorphisms of the graph Γ F (f 2 , f 3 ), where F is a field; see, e.g., [14] (p. 3) or [11] (Proposition 2.2, p. 190) for proofs. First note that for a function f = f (X, Y ), we define…”

“…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.…”