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- Brian G. Kronenthal
- Finite Fields and Their Applications
- 2012

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in… (More)

- Brian G. Kronenthal, Felix Lazebnik
- Discrete Applied Mathematics
- 2016

Let F be a field. For a polynomial f ∈ F[x, y], we define a bipartite graph ΓF(f) with vertex partition P ∪ L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). It is known that the graph ΓF(xy ) has no cycles of length less than eight. The main result of this paper is that ΓF(xy ) is the… (More)

A generalized die is a list (x1; :::; xn) of integers. For integers n 1, a b and s let D(n; a; b; s) be the set of all dice with a x1 ::: xn b and P xi = s. Two dice X and Y are tied if the number of pairs (i; j) with xi < yj equals the number of pairs (i; j) with xi > yj . We prove the following: with one exception (unique up to isomorphism), if X 6= Y 2… (More)

In this paper, we prove a closed form for a sequence motivated by the search for new generalized quadrangles of odd order. We present two proofs: a direct proof to explain the closed form’s derivation and a shorter inductive argument. The sequence in question is derived from congruences that arise from applying the Hermite-Dickson criterion to a permutation… (More)

- A Godsil, Ellen T Thomason, +41 authors Misha Muzychuk
- 2017

Let F be a field of characteristic zero or of a positive odd characteristic p. For a polynomial f ∈ F[x, y], we define a graph ΓF(xy, f) to be a bipartite graph with vertex partition P ∪L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). If f = xy, the graph ΓF(xy, xy ) has the length of… (More)

- Brian G. Kronenthal, Wing Hong Tony Wong
- Discrete Applied Mathematics
- 2017

Let n be a positive integer, and let d = (d1, d2, . . . , dn) be an n-tuple of integers such that di ≥ 2 for all i. A hypertorus Q d n is a simple graph defined on the vertex set {(v1, v2, . . . , vn) : 0 ≤ vi ≤ di − 1 for all i}, and has edges between u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn) if and only if there exists a unique i such that |ui… (More)

We present a very tedious proof of the n = 4 case of the Tied Dice Theorem. Suppose X = (x1; x2; x3; x4); Y = (y1; y2; y3; y4) 2 D(4; a; b; s) are distinct, tied, non-balanced dice. Let denote the number of pairs (i; j) with xi = yj ; as X and Y are tied, there must be (16 )=2 pairs (xi; yj) with xi > yj , and (16 )=2 pairs (xi; yj) with xi < yj . For each… (More)

- Brian G. Kronenthal, Felix Lazebnik, —Boris Pasternak
- 2011

Undoubtedly, every reader has tried to clarify a notion in one source by consulting another, only to be frustrated that the presentations are inconsistent in vocabulary or notation. Recently, this happened to us in a study of conics. While reading Peter Cameron’s Combinatorics: Topics, Techniques, and Algorithms [6], we encountered a definition of a… (More)

If you are uncertain about the meaning of the terms “quadratic form” or “linear form”, definitions will appear in Section 2. Interestingly enough, this problem has a simple solution, and was answered several centuries ago. However, many people we talked to found it intriguing and were surprised they had not thought about or seen this problem before. A… (More)