Continuants and some decompositions into squares
- Authors: Delorme, Charles , Pineda-Villavicencio, Guillermo
- Date: 2015
- Type: Text , Journal article
- Relation: Integers Vol. 15, no. (2015), p. 1
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- Description: In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number
Quadratic form representations via generalized continuants
- Authors: Delorme, Charles , Pineda-Villavicencio, Guillermo
- Date: 2015
- Type: Text , Journal article
- Relation: Journal of Integer Sequences Vol. 18, no. 6 (2015), p. Article number 15.6.4
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- Description: H. J. S. Smith proved Fermat’s two-square theorem using the notion of palindromic continuants. In this paper we extend Smith’s approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations. © 2015 University of Waterloo. All rights reserved.
On graphs with cyclic defect or excess
- Authors: Delorme, Charles , Pineda-Villavicencio, Guillermo
- Date: 2010
- Type: Text , Journal article
- Relation: Electronic Journal of Combinatorics Vol. 17, no. 1 (2010), p.
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- Description: The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ε are called graphs with defect or excess ε, respectively. While Moore graphs (graphs with ε = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation Gd,k(A) = Jn +B (Gd,k(A) = Jn - B), where A denotes the adjacency matrix of the graph in question, n its order, Jn the n × n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and Gd,k(x) a polynomial with integer coefficients such that the matrix Gd,k(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d3/2) for the number of graphs of odd degree d ≥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d ≥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree ≥ 3 and cyclic defect or excess exists.
On bipartite graphs of defect 2
- Authors: Delorme, Charles , Jorgensen, Leif , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: European Journal of Combinatorics Vol. 30, no. 4 (2009), p. 798-808
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- Description: It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree
On bipartite graphs of diameter 3 and defect 2
- Authors: Delorme, Charles , Jorgensen, Leif , Miller, Mirka , Pineda-Villavicencio, Guillermo
- Date: 2009
- Type: Text , Journal article
- Relation: Journal of Graph Theory Vol. 61, no. 4 (2009), p. 271-288
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- Description: We consider bipartite graphs of degree A<2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (â–³,3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (â–³,3, -2) - graphs. The most general of these conditions is that either â–³ or â–³-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when â–³ = 6 and â–³ = 9, we prove the non-existence of the corresponding bipartite (â–³,3,-2)-graphs, thus establishing that there are no bipartite (â–³,3, -2)-graphs, for 5