Searcharxiv⌕ Search

arXiv subjects

Francesc Comellas

Publications and source records attributed to Francesc Comellas.

At least 19 recordsLinked to original sources
arXiv2026

Non-Isomorphic Optimal Cayley Graphs

We present eighty-nine optimal degree-diameter Cayley graphs that are non-isomorphic to previously known optimal Cayley graphs with the same degree and diameter. For each graph, we provide comprehensive data on its parameters and structural invariants. Comparison with known constructions reported by Marston Conder on the Combinatorics Wiki and in related works reveals significant differences in girth, algebraic connectivity, domination number, automorphism group structure, cycle distributions, distance-related properties, and other graph invariants. These findings show that optimal degree-diameter Cayley graphs are frequently non-unique. The new examples enlarge the catalogue of known optimal graphs and provide benchmarks for investigations in degree-diameter problems, algebraic graph theory, and extremal graph theory. Since some of the graphs were identified through randomized search techniques, the collection is not exhaustive.

math.CO↗
arXiv2026

Table of large graphs with given degree and diameter

We update the table of large undirected graphs with given degree and diameter with results obtained since the publication of the survey by M. Miller and J. Širáň in the {\em Electronic Journal of Combinatorics} (Dynamic Survey DS14, 2nd. edition. May 2013).

math.CO↗
arXiv2015

Farey Graphs as Models for Complex Networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.

cond-mat.stat-mech↗
arXiv2015

The number and degree distribution of spanning trees in the Tower of Hanoi graph

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the num- ber and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.

cs.DM↗
arXiv2015

The normalized Laplacian spectrum of subdivisions of a graph

Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.

math.CO↗
arXiv2015

On the spectrum of the normalized Laplacian of iterated triangulations of graphs

The eigenvalues of the normalized Laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects, specifically in relation to random walks. In this paper we determine the spectra of the normalized Laplacian of iterated triangulations of a generic simple connected graph. As an application, we also find closed-forms for their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.

math.SP↗
arXiv2014

The Number of Spanning Trees in Apollonian Networks

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling and highly clustered, the study of their spanning trees is of particular relevance. Our results allow also the calculation of the spanning tree entropy of Apollonian networks, which then we compare with those of other graphs with the same average degree.

math.CO↗
arXiv2014

The Number of Spanning Trees of an Infinite Family of Outerplanar, Small-World and Self-Similar Graphs

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.

math.CO↗
arXiv2009

Planar unclustered scale-free graphs as models for technological and biological networks

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular several families of technological and biological networks, and in the design of new practical communication algorithms in relation to their dynamical processes. They can also help understanding the underlying mechanisms that have produced their particular structure.

physics.soc-ph↗
arXiv2009

Planar unclustered graphs to model technological and biological networks

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular technological and biological networks.

cond-mat.stat-mech↗
arXiv2008

Self-similar planar graphs as models for complex networks

In this paper we introduce a family of planar, modular and self-similar graphs which have small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated to complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees.

physics.soc-ph↗
arXiv2006

Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.

physics.soc-ph↗
arXiv2005

Evolving small-world networks with geographical attachment preference

We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyze several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as the small-world effect and a high clustering.

cond-mat.stat-mech↗
arXiv2005

Synchronous and Asynchronous Recursive Random Scale-Free Nets

We investigate the differences between scale-free recursive nets constructed by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus an asynchronous, random sequential updating rule (e.g., random Apollonian nets). We show that the dramatic discrepancies observed recently for the degree exponent in these two cases result from a biased choice of the units to be updated sequentially in the asynchronous version.

cond-mat.stat-mech↗
arXiv2005

High Dimensional Apollonian Networks

We propose a simple algorithm which produces high dimensional Apollonian networks with both small-world and scale-free characteristics. We derive analytical expressions for the degree distribution, the clustering coefficient and the diameter of the networks, which are determined by their dimension.

cond-mat.other↗
arXiv2005

High dimensional random Apollonian networks

We propose a simple algorithm which produces a new category of networks, high dimensional random Apollonian networks, with small-world and scale-free characteristics. We derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network. The values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks. We prove also analitically that the average path length of the networks increases at most logarithmically with the number of vertices.

cond-mat.other↗
arXiv2004

Recursive graphs with small-world scale-free properties

We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. We determine relevant characteristics of those graphs: the diameter, degree distribution, and clustering parameter. The graphs have also an interesting recursive property, and generalize recent constructions with fixed degree distributions.

cond-mat.stat-mech↗