### On the non-existence of even degree graphs of diameter 2 and defect 2

**Authors:**Miller; Mirka , Nguyen, Minh Hoang , Pineda-Villavicencio, Guillermo**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 18th International Workshop on Combinatorial Algorithms, IWOCA 2007, Rafferty's Resort, Lake Macquarie, New South Wales : 5th-9th November 2007**Full Text:****Description:**Using eigenvalue analysis, it was shown by Erdos et al. 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, when d is even, and we conjecture that they do not exist for any even d greater than 4.**Description:**2003007893

### 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**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 the nonexistence of graphs of diameter 2 and defect 2

**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

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

**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.

### On graphs of maximum degree 3 and defect 4

**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

### Fitting Voronoi diagrams to planar tesselations

**Authors:**Aloupis, Greg , Pérez-Rosés, Hebert , Pineda-Villavicencio, Guillermo , Taslakian, Perouz , Trinchet-Almaguer, Dannier**Date:**2013**Type:**Text , Conference paper**Relation:**Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Vol. 8288 LNCS, p. 349-361**Full Text:****Reviewed:****Description:**Given a tesselation of the plane, defined by a planar straight-line graph G, we want to find a minimal set S of points in the plane, such that the Voronoi diagram associated with S 'fits' G. This is the Generalized Inverse Voronoi Problem (GIVP), defined in [12] and rediscovered recently in [3]. Here we give an algorithm that solves this problem with a number of points that is linear in the size of G, assuming that the smallest angle in G is constant. © 2013 Springer-Verlag.

### 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**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

### The degree-diameter problem for sparse graph classes

**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

### Measuring Surveillance in Online Advertising: A Big Data Approach

**Authors:**Herps, Aaron , Watters, Paul , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Conference paper**Relation:**Proceedings - 4th Cybercrime and Trustworthy Computing Workshop, CTC 2013 p. 30-35**Full Text:**false**Reviewed:****Description:**There is an increasing public and policy awareness that tracking cookies are being used to support behavioral advertising, but the extent to which tracking is occurring is not clear. The extent of tracking could have implications for the enforceability of legislative responses to the sharing of personal data, including the Privacy Act 1988 (Cth). In this paper, we develop a methodology for determining the prevalence of tracking cookies, and report the results for a sample of the 50 most visited sites by Australians. We find that the use of tracking cookies is endemic, but that distinct clusters of tracking can be identified across categories including search, pornography and social networking. The implications of the work in relation to privacy are discussed.

### 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.**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.

### The maximum degree & diameter-bounded subgraph and its applications

**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.

### The excess degree of a polytope

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Discrete Mathematics Vol. 32, no. 3 (2018), p. 2011-2046**Full Text:****Reviewed:****Description:**We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d \not = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.

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

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

**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.

### New largest graphs of diameter 6. (Extended Abstract)

**Authors:**Pineda-Villavicencio, Guillermo , Gomez, Jose , Miller, Mirka , Pérez-Rosés, Hebert**Date:**2006**Type:**Text , Journal article**Relation:**Electronic Notes in Discrete Mathematics Vol. 24, no. (2006), p. 153-160**Full Text:****Reviewed:****Description:**In the pursuit of obtaining largest graphs of given degree and diameter, many construction techniques have arisen. Compounding of graphs is one such technique. In this paper, by means of the compounding of complete graphs into the bipartite Moore graph of diameter 6, we obtain two families of (**Description:**C1

### Topology of interconnection networks with given degree and diameter

**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:**

### On graphs of defect at most 2

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 159, no. 13 (2011), p. 1331-1344**Full Text:****Reviewed:****Description:**In this paper we consider the degree/diameter problem, namely, given natural numbers Î”<2 and D<1, find the maximum number N(Î”,D) of vertices in a graph of maximum degree Î” and diameter D. In this context, the Moore bound M(Î”,D) represents an upper bound for N(Î”,D). Graphs of maximum degree Î”, diameter D and order M(Î”,D), called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Î”<2, diameter D<1 and order M(Î”,D)- with small >0, that is, (Î”,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Î”=2. When >1, (Î”,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-PurÃ³n and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Î”,D,-2)-graph with Î”<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Î”,D,-2)-graphs with even Î”<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0â‰¤â‰¤2. Such a catalogue is only the second census of (Î”,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0â‰¤â‰¤2 JÃ¸rgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Î”,D,-2)-graphs with odd Î”<5 and D<4, and the non-existence of (Î”,D,-2)-graphs with odd Î”<5 and D<5 such that Î”â‰¡0,2(modD). Finally, we conjecture that there are no (Î”,D,-2)-graphs with Î”<4 and D<4, and comment on some implications of our results for the upper bounds of N(Î”,D). Â© 2011 Elsevier B.V. All rights reserved.

### The Maximum Degree & Diameter-Bounded Subgraph and its Applications

**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. , no. (2012), p. 1-20**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. © 2012 Springer Science+Business Media B.V.

### Topology of interconnection networks with given degree and diameter

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