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9Miller, Mirka
6Ugon, Julien
5Delorme, Charles
5Pérez-Rosés, Hebert
5Yost, David
4Feria-Purón, Ramiro
3Nevo, Eran
3Watters, Paul
2Dekker, Anthony
2Jorgensen, Leif
2Nguyen, Minh Hoang
2Wood, David
1Aloupis, Greg
1Atchison, Rory
1Bui, Hoa
1Doolittle, Joseph
1Gomez, Jose
1Gómez, José
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150101 Pure Mathematics
80802 Computation Theory and Mathematics
7Degree/diameter problem
6Defect
50103 Numerical and Computational Mathematics
4Moore bound
4Moore graphs
30102 Applied Mathematics
3Bipartite Moore graphs
2Bipartite Moore bound
2Botnets
2Compounding of graphs
2Degree-diameter problem
2Degree/diameter problem for bipartite graphs
2F-vector
2Mesh
2Moore bipartite bound
2Moore graph
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The maximum degree & diameter-bounded subgraph and its applications

- Dekker, Anthony, Pérez-Rosés, Hebert, Pineda-Villavicencio, Guillermo, Watters, Paul

**Authors:**Dekker, Anthony , Pérez-Rosés, Hebert , Pineda-Villavicencio, Guillermo , Watters, Paul**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Mathematical Modelling and Algorithms Vol. 11, no. 3 (2012), p. 249-268**Full Text:**false**Reviewed:****Description:**We introduce the problem of finding the largest subgraph of a given weighted undirected graph (host graph), subject to constraints on the maximum degree and the diameter. We discuss some applications in security, network design and parallel processing, and in connection with the latter we derive some bounds for the order of the largest subgraph in host graphs of practical interest: the mesh and the hypercube. We also present a heuristic strategy to solve the problem, and we prove an approximation ratio for the algorithm. Finally, we provide some experimental results with a variety of host networks, which show that the algorithm performs better in practice than the prediction provided by our theoretical approximation ratio.

On the nonexistence of graphs of diameter 2 and defect 2

- Miller, Mirka, Nguyen, Minh Hoang, Pineda-Villavicencio, Guillermo

**Authors:**Miller, Mirka , Nguyen, Minh Hoang , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**The Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 71, no. (2009), p. 5-20**Full Text:**false**Reviewed:****Description:**In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree d and d² + 1 vertices), and found that such graphs exist only for d = 2; 3; 7 and possibly 57. In 1980, Erdös et al., using eigenvalue analysis, showed that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d² vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d² - 1 vertices do not exist for most values of d with d ≥ 6, and conjecture that they do not exist for any d ≥ 6.**Description:**2003007893

Almost simplicial polytopes : the lower and upper bound theorems

- Nevo, Eran, Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2020**Type:**Text , Journal article**Relation:**Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2020**Type:**Text , Journal article**Relation:**Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.

The degree-diameter problem for sparse graph classes

- Pineda-Villavicencio, Guillermo, Wood, David

**Authors:**Pineda-Villavicencio, Guillermo , Wood, David**Date:**2015**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 22, no. 2 (2015), p. 1-20**Full Text:**false**Reviewed:****Description:**The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree ∆ and diameter k. For fixed k, the answer is

Connectivity of cubical polytopes

- Bui, Hoa, Pineda-Villavicencio, Guillermo, Ugon, Julien

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

Continuants and some decompositions into squares

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2015**Type:**Text , Journal article**Relation:**Integers Vol. 15, no. (2015), p. 1**Full Text:****Reviewed:****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

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2015**Type:**Text , Journal article**Relation:**Integers Vol. 15, no. (2015), p. 1**Full Text:****Reviewed:****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

Constructions of large graphs on surfaces

- Feria-Purón, Ramiro, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Pineda-Villavicencio, Guillermo**Date:**2014**Type:**Text , Journal article**Relation:**Graphs and Combinatorics Vol. 30, no. 4 (2014), p. 895-908**Full Text:**false**Reviewed:****Description:**We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface

Complete catalogue of graphs of maximum degree 3 and defect at most 4

- Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 157, no. 13 (2009), p. 2983-2996**Full Text:****Reviewed:****Description:**We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.

**Authors:**Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 157, no. 13 (2009), p. 2983-2996**Full Text:****Reviewed:****Description:**We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.

On graphs of maximum degree 3 and defect 4

- Pineda-Villavicencio, Guillermo, Miller, Mirka

**Authors:**Pineda-Villavicencio, Guillermo , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of combinatorial mathematics and combinatorial computing Vol. 65, no. (May 2008), p. 25-31**Full Text:**false**Reviewed:****Description:**It is well known that apart from the Petersen graph there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and

On the reconstruction of polytopes

- Doolittle, Joseph, Nevo, Eran, Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**Authors:**Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

**Authors:**Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Quadratic form representations via generalized continuants

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**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**Full Text:**false**Reviewed:****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

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 17, no. 1 (2010), p.**Full Text:****Reviewed:****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 Îµ are called graphs with defect or excess Îµ, respectively. While Moore graphs (graphs with Îµ = 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 G_{d,k}(A) = J_{n}+B (G_{d,k}(A) = J_{n}- B), where A denotes the adjacency matrix of the graph in question, n its order, J_{n}the n Ã— n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and G_{d,k}(x) a polynomial with integer coefficients such that the matrix G_{d,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 d^{3/2}) for the number of graphs of odd degree d â‰¥ 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 â‰¥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the MÃ¶bius ladder on 8 vertices, no non-trivial graph of any degree â‰¥ 3 and cyclic defect or excess exists.

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 17, no. 1 (2010), p.**Full Text:****Reviewed:****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 Îµ are called graphs with defect or excess Îµ, respectively. While Moore graphs (graphs with Îµ = 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 G_{d,k}(A) = J_{n}+B (G_{d,k}(A) = J_{n}- B), where A denotes the adjacency matrix of the graph in question, n its order, J_{n}the n Ã— n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and G_{d,k}(x) a polynomial with integer coefficients such that the matrix G_{d,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 d^{3/2}) for the number of graphs of odd degree d â‰¥ 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 â‰¥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the MÃ¶bius ladder on 8 vertices, no non-trivial graph of any degree â‰¥ 3 and cyclic defect or excess exists.

On bipartite graphs of defect 2

- Delorme, Charles, Jorgensen, Leif, Miller, Mirka, Pineda-Villavicencio, Guillermo

**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**Full Text:****Reviewed:****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

**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**Full Text:****Reviewed:****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 large bipartite graphs of diameter 3

- Feria-Purón, Ramiro, Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

Topology of interconnection networks with given degree and diameter

- Pineda-Villavicencio, Guillermo

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**Doctor of Philosophy

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**Doctor of Philosophy

Lower bound theorems for general polytopes

- Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45**Full Text:****Reviewed:****Description:**For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45**Full Text:****Reviewed:****Description:**For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

Non-existence of bipartite graphs of diameter at least 4 and defect 2

- Pineda-Villavicencio, Guillermo

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Algebraic Combinatorics Vol. 34, no. 2 (2011), p. 163-182**Full Text:****Reviewed:****Description:**The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Î” and diameter D. Bipartite graphs of maximum degree Î”, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Î” such that Î”-1 is a prime power. The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree Î”, diameter D, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite (Î”,D,-2)-graphs. For Î”â‰¥3 bipartite (Î”,2,-2)-graphs are the complete bipartite graphs with partite sets of orders Î” and Î”-2. In this paper we consider bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥3. Some necessary conditions for the existence of bipartite (Î”,3,-2)-graphs for Î”â‰¥3 are already known, as well as the non-existence of bipartite (Î”,D,-2)-graphs with Î”â‰¥3 and D=4,5,6,8. Furthermore, it had been conjectured that bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥4 do not exist. Here, using graph spectra techniques, we completely settle this conjecture by proving the non-existence of bipartite (Î”,D,-2)-graphs for all Î”â‰¥3 and all Dâ‰¥6. Â© 2010 Springer Science+Business Media, LLC.

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Algebraic Combinatorics Vol. 34, no. 2 (2011), p. 163-182**Full Text:****Reviewed:****Description:**The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Î” and diameter D. Bipartite graphs of maximum degree Î”, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Î” such that Î”-1 is a prime power. The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree Î”, diameter D, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite (Î”,D,-2)-graphs. For Î”â‰¥3 bipartite (Î”,2,-2)-graphs are the complete bipartite graphs with partite sets of orders Î” and Î”-2. In this paper we consider bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥3. Some necessary conditions for the existence of bipartite (Î”,3,-2)-graphs for Î”â‰¥3 are already known, as well as the non-existence of bipartite (Î”,D,-2)-graphs with Î”â‰¥3 and D=4,5,6,8. Furthermore, it had been conjectured that bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥4 do not exist. Here, using graph spectra techniques, we completely settle this conjecture by proving the non-existence of bipartite (Î”,D,-2)-graphs for all Î”â‰¥3 and all Dâ‰¥6. Â© 2010 Springer Science+Business Media, LLC.

Using radar plots for performance benchmarking at patient and hospital levels using an Australian orthopaedics dataset

- Morales-Silva, Daniel, McPherson, Cameron, Pineda-Villavicencio, Guillermo, Atchison, Rory

**Authors:**Morales-Silva, Daniel , McPherson, Cameron , Pineda-Villavicencio, Guillermo , Atchison, Rory**Date:**2020**Type:**Text , Journal article**Relation:**Health Informatics Journal Vol. 26, no. 3 (2020), p. 2119-2137**Full Text:****Reviewed:****Description:**This study will highlight the diagnostic potential that radar plots display for reporting on performance benchmarking from patient admissions to hospital for surgical procedures. Two drawbacks of radar plots – the presence of missing information and ordering of indicators – are addressed. Ten different orthopaedic surgery procedures were considered in this study. Moreover, twelve outcome indicators were provided for each of the 10 surgeries of interest. These indicators were displayed using a radar plot, which we call a scorecard. At the hospital level, we propose a facile process by which to consolidate our 10 scorecards into one. We addressed the ordering of indicators in our scorecards by considering the national median of the indicators as a benchmark. Furthermore, our the consolidated scorecard facilitates concise visualisation and dissemination of complex data. It also enables the classification of providers into potential low and high performers that warrant further investigation. In conclusion, radar plots provide a clear and effective comparative tool for discerning multiple outcome indicators against the benchmarks of patient admission. A case study between two top and bottom performers on a consolidated scorecard (at hospital level) showed that medical provider charges varied more than other outcome indicators. © The Author(s) 2020.

**Authors:**Morales-Silva, Daniel , McPherson, Cameron , Pineda-Villavicencio, Guillermo , Atchison, Rory**Date:**2020**Type:**Text , Journal article**Relation:**Health Informatics Journal Vol. 26, no. 3 (2020), p. 2119-2137**Full Text:****Reviewed:****Description:**This study will highlight the diagnostic potential that radar plots display for reporting on performance benchmarking from patient admissions to hospital for surgical procedures. Two drawbacks of radar plots – the presence of missing information and ordering of indicators – are addressed. Ten different orthopaedic surgery procedures were considered in this study. Moreover, twelve outcome indicators were provided for each of the 10 surgeries of interest. These indicators were displayed using a radar plot, which we call a scorecard. At the hospital level, we propose a facile process by which to consolidate our 10 scorecards into one. We addressed the ordering of indicators in our scorecards by considering the national median of the indicators as a benchmark. Furthermore, our the consolidated scorecard facilitates concise visualisation and dissemination of complex data. It also enables the classification of providers into potential low and high performers that warrant further investigation. In conclusion, radar plots provide a clear and effective comparative tool for discerning multiple outcome indicators against the benchmarks of patient admission. A case study between two top and bottom performers on a consolidated scorecard (at hospital level) showed that medical provider charges varied more than other outcome indicators. © The Author(s) 2020.

On bipartite graphs of defect at most 4

- Feria-Purón, Ramiro, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol.160, no.1-2 (2011), p.140-154**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Î” â‰¥ 2 and D â‰¥ 2, find the maximum number Nb (Î”, D) of vertices in a bipartite graph of maximum degree Î” and diameter D. In this context, the Moore bipartite bound Mb (Î”, D) represents an upper bound for Nb (Î”, D). Bipartite graphs of maximum degree Î”, diameter D and order Mb (Î”, D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Î” â‰¥ 2, diameter D â‰¥ 2 and order Mb (Î”, D) - Îµ{lunate} with small Îµ{lunate} > 0, that is, bipartite (Î”, D, - Îµ{lunate})-graphs. The parameter Îµ{lunate} is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if Î” â‰¥ 3 and D â‰¥ 3, they may only exist for D = 3. However, when Îµ{lunate} > 2 bipartite (Î”, D, - Îµ{lunate})-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (Î”, D, - 4)-graphs; the complete catalogue of bipartite (3, D, - Îµ{lunate})-graphs with D â‰¥ 2 and 0 â‰¤ Îµ{lunate} â‰¤ 4; the complete catalogue of bipartite (Î”, D, - Îµ{lunate})-graphs with Î” â‰¥ 2, 5 â‰¤ D â‰¤ 187 (D â‰ 6) and 0 â‰¤ Îµ{lunate} â‰¤ 4; a proof of the non-existence of all bipartite (Î”, D, - 4)-graphs with Î” â‰¥ 3 and odd D â‰¥ 5. Finally, we conjecture that there are no bipartite graphs of defect 4 for Î” â‰¥ 3 and D â‰¥ 5, and comment on some implications of our results for the upper bounds of Nb (Î”, D). Â© 2011 Elsevier B.V. All rights reserved.

**Authors:**Feria-Purón, Ramiro , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol.160, no.1-2 (2011), p.140-154**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Î” â‰¥ 2 and D â‰¥ 2, find the maximum number Nb (Î”, D) of vertices in a bipartite graph of maximum degree Î” and diameter D. In this context, the Moore bipartite bound Mb (Î”, D) represents an upper bound for Nb (Î”, D). Bipartite graphs of maximum degree Î”, diameter D and order Mb (Î”, D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Î” â‰¥ 2, diameter D â‰¥ 2 and order Mb (Î”, D) - Îµ{lunate} with small Îµ{lunate} > 0, that is, bipartite (Î”, D, - Îµ{lunate})-graphs. The parameter Îµ{lunate} is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if Î” â‰¥ 3 and D â‰¥ 3, they may only exist for D = 3. However, when Îµ{lunate} > 2 bipartite (Î”, D, - Îµ{lunate})-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (Î”, D, - 4)-graphs; the complete catalogue of bipartite (3, D, - Îµ{lunate})-graphs with D â‰¥ 2 and 0 â‰¤ Îµ{lunate} â‰¤ 4; the complete catalogue of bipartite (Î”, D, - Îµ{lunate})-graphs with Î” â‰¥ 2, 5 â‰¤ D â‰¤ 187 (D â‰ 6) and 0 â‰¤ Îµ{lunate} â‰¤ 4; a proof of the non-existence of all bipartite (Î”, D, - 4)-graphs with Î” â‰¥ 3 and odd D â‰¥ 5. Finally, we conjecture that there are no bipartite graphs of defect 4 for Î” â‰¥ 3 and D â‰¥ 5, and comment on some implications of our results for the upper bounds of Nb (Î”, D). Â© 2011 Elsevier B.V. All rights reserved.

Topology of interconnection networks with given degree and diameter

- Pineda-Villavicencio, Guillermo

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 81, no. 2 (2010), p. 350-352**Full Text:****Reviewed:**

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 81, no. 2 (2010), p. 350-352**Full Text:****Reviewed:**

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