Guillermo Pineda-Villavicencio

The Maximum Degree & Diameter-Bounded Subgraph and its Applications

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 bipartite graphs of defect at most 4

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers @D>=2 and D>=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 Moore bipartite bound M^b(@D,D) represents an upper bound for N^b(@D,D). Bipartite graphs of maximum degree @D, diameter D and order M^b(@D,D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree @D>=2, diameter D>=2 and order M^b(@D,D)-@e with small @e>0, that is, bipartite (@D,D,-@e)-graphs. The parameter @e 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 @D>=3 and D>=3, they may only exist for D=3. However, when @e>2 bipartite (@D,D,-@e)-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (@D,D,-4)-graphs; the complete catalogue of bipartite (3,D,-@e)-graphs with D>=2 and 0@?@e@?4; the complete catalogue of bipartite (@D,D,-@e)-graphs with @D>=2, 5@?D@?187 (D 6) and 0@?@e@?4; a proof of the non-existence of all bipartite (@D,D,-4)-graphs with @D>=3 and odd D>=5. Finally, we conjecture that there are no bipartite graphs of defect 4 for @D>=3 and D>=5, and comment on some implications of our results for the upper bounds of N^b(@D,D).

On bipartite graphs of defect 2

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 Δ≥2, diameter D≥2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (Δ,D,-2)-graphs. We find that the eigenvalues other than ±Δ of such graphs are the roots of the polynomials HD-1(x)±1, where HD-1(x) is the Dickson polynomial of the second kind with parameter Δ-1 and degree D-1. For any diameter, we prove that the irreducibility over the field of rational numbers of the polynomial HD-1(x)-1 provides a sufficient condition for the non-existence of bipartite (Δ,D,-2)-graphs for Δ≥3 and D≥4. Then, by checking the irreducibility of these polynomials, we prove the non-existence of bipartite (Δ,D,-2)-graphs for all Δ≥3 and D∈{4,6,8}. For odd diameters, we develop an approach that allows us to consider only one partite set of the graph in order to study the non-existence of the graph. Based on this, we prove the non-existence of bipartite (Δ,5,-2)-graphs for all Δ≥3. Finally, we conjecture that there are no bipartite (Δ,D,-2)-graphs for Δ≥3 and D≥4.

On graphs of defect at most 2

Abstract In this paper we consider the degree/diameter problem, namely, given natural numbers Δ ≥ 2 and D ≥ 1 , find the maximum number N ( Δ , D ) of vertices in a graph of maximum degree Δ and diameter D . In this context, the Moore bound M ( Δ , D ) represents an upper bound for N ( Δ , D ) . Graphs of maximum degree Δ , diameter D and order M ( Δ , D ) , called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Δ ≥ 2 , diameter D ≥ 1 and order M ( Δ , D ) − ϵ with small ϵ > 0 , that is, ( Δ , D , − ϵ ) -graphs. The parameter ϵ is called the defect. Graphs of defect 1 exist only for Δ = 2 . When ϵ > 1 , ( Δ , D , − ϵ ) -graphs represent a wide unexplored area. This paper focuses on graphs of defect 2 . Building on the approaches developed in Feria-Puron and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ( Δ , D , − 2 ) -graph with Δ ≥ 4 and D ≥ 4 is 2 D . Second, and most important, we prove the non-existence of ( Δ , D , − 2 ) -graphs with even Δ ≥ 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 , − ϵ ) -graphs with D ≥ 2 and 0 ≤ ϵ ≤ 2 . Such a catalogue is only the second census of ( Δ , D , − 2 ) -graphs known at present, the first being that of ( 3 , D , − ϵ ) -graphs with D ≥ 2 and 0 ≤ ϵ ≤ 2 Jorgensen (1992) [14] . Other results of this paper include necessary conditions for the existence of ( Δ , D , − 2 ) -graphs with odd Δ ≥ 5 and D ≥ 4 , and the non-existence of ( Δ , D , − 2 ) -graphs with odd Δ ≥ 5 and D ≥ 5 such that Δ ≡ 0 , 2 ( mod D ) . Finally, we conjecture that there are no ( Δ , D , − 2 ) -graphs with Δ ≥ 4 and D ≥ 4 , and comment on some implications of our results for the upper bounds of N ( Δ , D ) .

Fitting Voronoi diagrams to planar tesselations

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.

On large bipartite graphs of diameter 3

Abstract We consider the bipartite version of the degree/diameter problem , namely, given natural numbers d ≥ 2 and D ≥ 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 M b ( d , D ) represents a general upper bound for N b ( d , D ) . Bipartite graphs of order M b ( d , D ) are very rare, and determining N b ( d , D ) still remains an open problem for most ( d , D ) pairs. This paper is a follow-up of our earlier paper (Feria-Puron 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 N b ( 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 N b ( 11 , 3 ) .

New Benchmarks for Large-Scale Networks with Given Maximum Degree and Diameter

Large-scale networks have become ubiquitous elements of our society. Modern social networks, supported by communication and travel technology, have grown in size and complexity to unprecedented scales. Computer networks, such as the Internet, have a fundamental impact on commerce, politics and culture. The study of networks is also central in biology, chemistry and other natural sciences. Unifying aspects of these networks are a small maximum degree and a small diameter, which are also shared by many network models, such as small-world networks. Graph theoretical methodologies can be instrumental in the challenging task of predicting, constructing and studying the properties of large-scale networks. This task is now necessitated by the vulnerability of large networks to phenomena such as cross-continental spread of disease and botnets (networks of malware). In this article, we produce the new largest known networks of maximum degree 17 ≤ Δ ≤ 20 and diameter 2 ≤ D ≤ 10, using a wide range of techniques and concepts, such as graph compounding, vertex duplication, Kronecker product, polarity graphs and voltage graphs. In this way, we provide new benchmarks for networks with given maximum degree and diameter, and a complete overview of state-of-the-art methodology that can be used to construct such networks.

On the maximum order of graphs embedded in surfaces

The maximum number of vertices in a graph of maximum degree Δ ? 3 and fixed diameter k ? 2 is upper bounded by ( 1 + o ( 1 ) ) ( Δ - 1 ) k . If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of ( 2 + o ( 1 ) ) ( Δ - 1 ) ? k / 2 ? for a fixed k. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus g behave like trees, in the sense that, for large Δ, such graphs have orders bounded from above by { c ( g + 1 ) ( Δ - 1 ) ? k / 2 ? if? k ?is even c ( g 3 / 2 + 1 ) ( Δ - 1 ) ? k / 2 ? if? k ?is odd , where c is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter k. With respect to lower bounds, we construct graphs of Euler genus g, odd diameter k, and order c ( g + 1 ) ( Δ - 1 ) ? k / 2 ? for some absolute constant c 0 . Our results answer in the negative a question of Miller and Siraň (2005).

Constructions of Large Graphs on Surfaces

AbstractWe consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers Δ and k, determine the maximum order N(Δ,k,Σ) of a graph embeddable in Σ with maximum degree Δ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface Σ of Euler genus g and an odd diameter k, the current best asymptotic lower bound for N(Δ,k,Σ) is given by