On the girth of two-dimensional real algebraically defined graphs

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

“…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 discussion motivates the following question about girth eight graphs in the three-dimensional real case.…”

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

“…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. A number of questions related to connectivity, diameter, and isomorphisms of similarly constructed directed graphs, as well as a peculiar result on the number of roots of certain polynomials in finite fields, were considered in Kodess [6], Kodess and Lazebnik [7,8], Kodess, Lazebnik, Smith, and Sporre [9], and Coulter, De Winter, Kodess, and Lazebnik [1].…”

“…In this paper, as in [4], we study undirected graphs over the real numbers; but here, we examine the three-dimensional case. Our main result is the following classification of all such monomial graphs (the semicolons indicate logical conjunctions):…”

On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

“…In addition, Kronenthal and Lazebnik [17] and Kronenthal, Lazebnik, and Williford [18] 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 Cheng, Tang, and Xu [2]. Moreover, Kodess, Kronenthal, Manzano-Ruiz, and Noe [12] classified monomial graphs over the real numbers, and Ganger, Golden, Kronenthal, and Lyons [6] studied a two-dimensional analogue. A number of questions related to connectivity, diameter, and isomorphisms of similarly constructed directed graphs, as well as a peculiar result on the number of roots of certain polynomials in finite fields, were considered by Kodess [11], Kodess and Lazebnik [13,14], Kodess, Lazebnik, Smith, and Sporre [15], and Coulter, De Winter, Kodess, and Lazebnik [3].…”