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Let F q be a finite field, where q = p e for some odd prime p and integer e ≥ 1. proved in [6] that if p ≥ 5 and e = 2 a 3 b for integers a, b ≥ 0, then all monomial graphs G q (f, g) of girth at least eight are isomorphic to G q (xy, xy 2), an induced subgraph of the point-line incidence graph of a classical generalized quadrangle of order q. In this… (More)

Let F be a field. For a polynomial f ∈ F[x, y], we define a bipartite graph Γ F (f) with vertex partition

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)

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 Γ F (xy, f) to be a bipartite graph with vertex partition P ∪L, P = F 3 = 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 2 , the graph Γ F (xy, xy 2) has the… (More)

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The following corrections and comments resulted from questions that arose while carefully reading [1], as well as discussions with Felix Lazebnik in an attempt to answer these questions. I have also corrected several typographical errors. This document was written with the purpose of clarifying certain elements of the paper. • Page 3: Correct " Thge " to "… (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)

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