Nacho López

Non existence of some mixed Moore graphs of diameter 2 using SAT

Mixed graphs with maximum number of vertices regarding to a given maximum degree and given diameter are known as mixed Moore graphs. In this paper we model the problem of the existence of mixed Moore graphs of diameter 2 through the Boolean satisfiability problem. As a consequence, we prove the non existence of mixed Moore graphs of order 40, 54 and 84.

Mixed Moore Cayley graphs

Abstract Cayley graphs are well known objects with interesting properties, also in the context of Moore graphs and digraphs. In 1978 Bosak extended Moores property to the mixed setting (with arcs allowed along with edges), in the so called mixed Moore graphs: those having a unique trail between pairs of vertices at a distance smaller than or equal to the diameter. In this paper we adapt Cayleys construction to mixed graphs and we show certain mixed Moore graphs are Cayley while some other cannot be Cayley.

Properties of mixed Moore graphs of directed degree one

Mixed graphs of order n such that for any pair of vertices there is a unique trail of length at most k between them are known as mixed Moore graphs. These extremal graphs may only exist for diameter k = 2 and certain (infinitely many) values of n . In this paper we give some properties of mixed Moore graphs of directed degree one and reduce their existence to the existence of some (undirected) strongly regular graphs.

Triangle randomization for social network data anonymization

In order to protect privacy of social network participants, network graph data should be anonymised prior to its release. Most proposals in the literature aim to achieve k -anonymity under specific assumptions about the background information available to the attacker. Our method is based on randomizing the location of the triangles in the graph. We show that this simple method preserves the main structural parameters of the graph to a high extent, while providing a high re-identification confusion.

Radial Moore graphs of radius three

The maximum number of vertices in a graph of specified degree and diameter cannot exceed the Moore bound. Graphs achieving this bound are called Moore graphs. Because Moore graphs are so rare, researchers have considered various relaxations of the Moore graph constraints. Since the diameter of a Moore graph is equal to its radius, one can consider graphs in which the condition on the diameter is relaxed, by one, while the condition on the radius is maintained. Such graphs are called radial Moore graphs. It has previously been shown that radial Moore graphs exist for all degrees when the radius is two. In this paper, we extend this result to radius three. We also construct examples that settle the existence question for a few new cases, and summarize the state of knowledge on the problem.

Ranking measures for radially Moore graphs

For graphs with maximum degree d and diameter k, an upper bound on the number of vertices in the graphs is provided by the well‐known Moore bound (denoted by Md,k). Graphs that achieve this bound (Moore graphs) are very rare, and determining how close one can come to the Moore bound has been a major topic in graph theory. Of particular note in this regard are the cage problem and the degree/diameter problem. In this article, we take a different approach and consider questions that arise when we fix the number of vertices in the graph at the Moore bound, but relax, by one, the diameter constraint on a subset of the vertices. In this context, regular graphs of degree d, radius k, diameter k + 1, and order equal to Md,k are called radially Moore graphs. We consider two specific questions. First, we consider the existence question (extending the work of Knor), and second, we consider some natural measures of how well a radially Moore graph approximates a Moore graph.

Privacy preserving release of blogosphere data in the presence of search engines

Users registered in a blogging platform and the subscriptions among them compose a social network with non-symmetric relations, whose data can be modeled as a directed graph. Release of such data for scientific analysis requires a pre-processing for ensuring no private information about people will be disclosed. The measures to be taken depend on the previous structural information a dishonest analyst is assumed to have. In this paper, the considered previous information is the sorting of blogs according to their PageRank relevance, which can be obtained by querying the blogging platform search engine. After analyzing the scenario, the n-rank confusion model is proposed. Experimental results show this model achieves a high privacy protection level while preserving the structural parameters of directed graph data to a high extent.

Degree/diameter Problem for Mixed Graphs

The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs.

The Degree/Diameter Problem for mixed abelian Cayley graphs

Abstract This paper investigates the upper bounds for the number of vertices in mixed abelian Cayley graphs with given degree and diameter. Additionally, in the case when the undirected degree is equal to one, we give a construction that provides a lower bound.

A variant of the McKay-Miller-Širáň construction for mixed graphs

Abstract The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, and Siraň in [McKay, B., M. Miller and J. Siraň, A note on large graphs of diameter two and given maximum degree, J Comb Theo Ser B 74 (1998), 110-118], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-Siraň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.