Miguel Angel Fiol

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ⩽ 2I and κ k = δ if D ⩽ 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.

On the extraconnectivity of graphs

Abstract Given a simple connected graph G , let κ ( n ) [ λ ( n )] be the minimum cardinality of a set of vertices [edges], if any, whose deletion disconnects G and every remaining component has more than n vertices. For instance, the usual connectivity and the superconnectivity of G correspond to κ (0) and κ (1), respectively. This paper gives sufficient conditions, relating the diameter of G with its girth, to assure optimum values of these conditional connectivities.

Extraconnectivity of graphs with large girth

Following Harary, the conditional connectivity (edge-connectivity) of a graph with respect to a given graph-theoretic property is the minimum cardinality of a set of vertices (edges), if any, whose deletion disconnects the graph and every remaining component has such a property. We study the case in which all these components are different from a tree whose order is not greater than n. For instance, the recently studied superconnectivity of a maximally connected graph corresponds to this conditional connectivity for n = 1. For other values of n, some sufficient conditions for a graph to have the maximum possible conditional connectivity are given.

Locally Pseudo-Distance-Regular Graphs

The concept of local pseudo-distance-regularity, introduced in this paper, can be thought of as a natural generalization of distance-regularity for non-regular graphs. Intuitively speaking, such a concept is related to the regularity of graph?when it is seen from a given vertex. The price to be paid for speaking about a kind of distance-regularity in the non-regular case seems to be locality. Thus, we find out that there are no genuine “global” pseudo-distance-regular graphs: when pseudo-distance-regularity is shared by all the vertices, the graph turns out to be distance-regular. Our main result is a characterization of locally pseudo-distance-regular graphs, in terms of the existence of the highest-degree member of a sequence of orthogonal polynomials. As a particular case, we obtain the following new characterization of distance-regular graphs: A graph?, with adjacency matrixA, is distance-regular if and only if?has spectrally maximum diameterD, all its vertices have eccentricityD, and the distance matrixADis a polynomial of degreeDinA.

Algebraic characterizations of distance-regular graphs

We survey some old and some new characterizations of distance-regular graphs, which depend on information retrieved from their adjacency matrix. In particular, it is shown that a regular graph with d + 1 distinct eigenvalues is distance-regular if and only if a numeric equality, involving only the spectrum of the graph and the numbers of vertices at distance d from each vertex, is satisfied.

From Local Adjacency Polynomials to Locally Pseudo-Distance-Regular Graphs

The local adjacency polynomials can be thought of as a generalization, for all graphs, of (the sums of ) the distance polynomials of distance-regular graphs. The term “local” here means that we “see” the graph from a given vertex, and it is the price we must pay for speaking of a kind of distance-regularity when the graph is not regular. It is shown that when the value at?(the maximum eigenvalue of the graph) of the local adjacency polynomials is large enough, then the eccentricity of the base vertex tends to be small. Moreover, when such a vertex is “tight” (that is, the value of a certain polynomial just fails to satisfy the condition) and fulfils certain additional extremality conditions, then all the polynomials attain their maximum possible values at?, and the graph turns out to be pseudo-distance-regular around the vertex. As a consequence of the above results, some new characterizations of distance-regular graphs are derived. For example, it is shown that a regular graph?withd+1 distinct eigenvalues is distance-regular if, and only if, the number of vertices at distancedfrom any given vertex is the value at?of the highest degree member of an orthogonal system of polynomials, which depend only on the spectrum of the graph.

Double commutative-step digraphs with minimum diameters

Abstract From a natural generalization to Z 2 of the concept of congruence, it is possible to define a family of 2-regular digraphs that we call double commutative-step digraphs. They turn out to be Cayley diagrams of Abelian groups generated by two elements, and can be represented by L-shaped tiles which tessellate the plane periodically. A double commutative-step digraph with n vertices is said to be tight if its diameter attains the lower bound lb (n)= 3n ˥−2 . In this paper, the tiles associated with tight double commutative-step digraphs are fully characterized. This allows us to construct such digraphs and also to find for which values of n they exist. In particular, we characterize a complete set of families of tight double fixed-step digraphs (also called circulant digraphs), generalizing some previously known results.

An efficient algorithm to find optimal double loop networks

The problem of finding optimal diameter double loop networks with a fixed number of vertices has been widely studied. In this work, we give an algorithmic solution of the problem by using a geometrical approach. Given a fixed number of vertices n, the general problem is to find “steps” s1, s2 ∈ Zn, such that the digraph G(n; s1, s2) with set of vertices V = Zn and adjacencies given by i → i + s1 (mod n) and i → i + s2 (mod n) has minimum diameter d(n). A lower bound of this diameter is known to be be lb(n)=⌈√3n⌉−2. So, given n, the algorithm has as outputs s1,s2 and the minimum integer κ = κ(n) such that d(n;s1,s2)=d(n)=lb(n)+κ The running time complexity of the algorithm is O(κ3)O(logn)-O(κ) is unknown but it is upper-bounded by O(4√n). Moreover, in most of the cases the algorithm also gives (as a by-product) an infinite family of digraphs with increasing order and diameter as above, to which the obtained digraph G(n; S1,S2) belongs.

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

On a Class of Polynomials and Its Relation with the Spectra and Diameters of Graphs

Let?1?2>?>?dbe points on the real line. For everyk=1, 2, ?, d, thek-alternatingpolynomialPkis the polynomial of degreekand norm ?Pk?∞=max1?l?d{|Pk(?l)|}?1 that attains maximum absolute value at any point???d, ?1]. Because of this optimality property, these polynomials may be thought of as the discrete version of the Chebychev polynomialsTkand, for particular values of the given points,Pkcoincides in fact with the “shifted”Tk. In general, however, those polynomials seem to bear a much more involved structure than Chebychev ones. Some basic properties of thePkare studied, and it is shown how to compute them in general. The results are then applied to the study of the relationship between the (standard or Laplacian) spectrum of a (not necessarily regular) graph or bipartite graph and its diameter, improving previous results.