Brian G. Kronenthal

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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)
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)
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)
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)