A Result on Polynomials Derived Via Graph Theory

Abstract

Graph theory is a comparatively young mathematical discipline. It is often hard to construct graphs that satisfy certain properties purely combinatorially, i.e., by taking a set of vertices and saying which vertex is connected to which. Often such areas of classical mathematics as number theory, geometry, or algebra are used for this, and the methods from the related areas are used to prove the properties of the obtained graphs. The examples are numerous, and many of them can be found in books and comprehensive survey articles. See, for example, Alon [2]; Babai and Frankl [5]; Biggs [6]; Füredi and Simonovits [12]; Brouwer and Haemers [8], and Alon and Spencer [3]. Here we wish to mention just a few such applications. The probabilistic method was used to prove the existence of certain graphs in Ramsey theory, and explicit constructions for these graphs are still unknown (see [3]). Constructions and analysis of Ramanujan graphs are often based on algebra and number theory. Methods of linear algebra are fundamental for studies of expanders and graphs with high degree of symmetry (see [5] and [8]). Lovász's proof [20] of a conjecture on the chromatic number of Kneser graphs made use of algebraic topology.Can the direction be reversed, i.e., can graph theory be used to obtain results in some classical areas of mathematics? Sometimes it can, but the number of such instances is much smaller. See, for example, Swan's proof of the Amitsur-Levitzki theorem [24], or a counterexample to Borsuk's conjecture by Kahn and Kalai [14] and related work by Bondarenko [7]. Extremal graph theory was used in probability by Katona [15], and in geometry and potential theory by Turán [25], and Erdős, Meir, Sos, and Turán [11]. For some applications of graph theory to linear algebra, see Doob [10]. A number of applications of graph theory to pure mathematics are mentioned in Lovász, Pyber,

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

“…A three-dimensional algebraically defined graph Γ R (f 2 (X, Y ), f 3 (X, Y )), or simply an algebraically defined graph, is a bipartite graph constructed using a ring R and two bivariate functions f 2 , f 3 : R 2 → R. Each partite set is a copy of R 3 , where vertices are labeled as (a, a 2 , a 3 ) in the first partite set and [x, x 2 , x 3 ] in the second. Two vertices are adjacent, denoted by (a, a 2 , a 3 ) ∼ [x, x 2 , x 3 ], if their coordinates satisfy the equations a i + x i = f i (a, x) for i ∈ {2, 3}.…”

“…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]. For more related results, see the survey paper by Lazebnik, Sun, and Wang [19].…”

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

“…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].…”