Cospectral digraphs from locally line digraphs

A digraph \G=(V,E) is a line digraph when every pair of vertices $u,v\in V$ have either equal or disjoint in-neighborhoods. When this condition only applies for vertices in a given subset (with at least two elements), we say that \G is a locally line digraph. In this paper we give a new method to obtain a digraph \G' cospectral with a given locally line digraph \G with diameter $D$, where the diameter $D'$ of \G' is in the interval $[D-1,D+1]$. In particular, when the method is applied to De Bruijn or Kautz digraphs, we obtain cospectral digraphs with the same algebraic properties that characterize the formers

Moments in graphs

Let $G$ be a connected graph with vertex set $V$ and a {\em weight function} $\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\em $\rho$-moment} of $G$ at vertex $u$ is defined to be M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot) stands for the distance function. Adding up all these numbers, we obtain the {\em $\rho$-moment of $G$}: M_G^{\rho}=\sum_{u\in V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\em Wiener index} $W(G)$, when $\rho(u)=1/2$ for every $u\in V$, and the {\em degree distance} $D'(G)$, obtained when $\rho(u)=\delta(u)$, the degree of vertex $u$. In this paper we derive some exact formulas for computing the $\rho$-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding $\rho$-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same $\rho$-moment for every $\rho$ (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product

Deterministic hierarchical networks

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and they are also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the systems modeled. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems

Grafs, amics i coneguts

Com és ben sabut, un graf és un objecte matemàtic que modelitza l'existència d'una certa relació entre parells d'elements d'un conjunt donat. Aleshores, és natural que molts dels primers resultats sobre grafs facin referència a relacions entre persones o grups de persones. En aquest article, comentem quatre resultats d'aquest tipus, els quals tenen relació amb diverses teories generals de grafs i les seves aplicacions: el lema de les encaixades de mans (relacionat amb la coloració de grafs i l'àlgebra booleana), un lema sobre els coneguts i desconeguts en una festa (amb la teoria de Ramsey), un lema sobre els amics en comú (amb la distància-regularitat i la teoria de codis) i el teorema de les noces de Hall (amb la connectivitat de les xarxes). Aquestes quatre àrees de la teoria de grafs, amb problemes sovint fàcils de plantejar però molt difícils de resoldre, s'han desenvolupat extensament i actualment són motiu de nombrosos treballs de recerca. Com a exemples de resultats i problemes representatius d'aquestes àrees, els quals són motiu de discussió en aquest treball que presentem, podem citar els següents: el teorema dels quatre colors (T4C), els nombres de Ramsey, els problemes d'existència de grafs distància-regulars i de codis completament regulars i, finalment, l'estudi de les propietats topològiques de les xarxes d'interconnexió.As is well known, a simple nondirected graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. It is therefore not surprising that, at the beginning, many of the results concerning graphs made reference to relationships between a group of people. In this expository article, we comment on four results of this kind, as representatives or as a source of inspiration for various general theories on graphs and their applications. In some cases, such as Hall’s marriage theorem, we also describe its relation to other topics of graph theory, as network connectivity

Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs $P(n)$ and to a recent known family of mixed graphs $\Gamma(d,n,r)$ (having edges with and without direction). As a byproduct, the existence of perfect codes in $P(n)$ allows us to give a lower bound for the multiplicity of its eigenvalue $-1$