José Luis Andres Yebra

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.

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.

Dense bipartite digraphs

For its implications in the design of interconnection networks, it is interesting to find (a) (di)graphs with given maximum (out-)degree d and diameter D that have large order; (b) (di)graphs of given order and maximum (out-)degree d that have small diameter. (Di)graphs of either type are often called dense. This paper considers the case of bipartite digraphs. For problem (a) it is shown that a Moore-like bound on the order of such digraphs can be (and in fact is) attained only when D ≤ 4. For D > 4 a construction is presented that yields a family of bipartite digraphs with order larger than (d4 — 1)/d4 times the above-mentioned bound. For problem (b) an appropriate lower bound is derived and a construction is presented that provides bipartite digraphs of any (even) order whose diameter does not exceed this lower bound in more than one.

Boundary graphs: the limit case of a spectral property

Abstract Recently, several results bounding the diameter of a regular graph from its eigenvalues have been presented. They admit the following unified presentation: Let λ 0 > λ 1 > … > λ d be the d + 1 distinct eigenvalues of a graph of order n and diameter D , and let P be a polynomial. Then, P ( λ 0 ) > ‖ P ‖ ∞ ( n −1) ⇒ D ⩽ dgr P , where where ‖ P ‖ ∞ = max 1 ⩽ i ⩽ d {| P ( λ i )|}. The best results are obtained when P = P k is the so-called k -alternating polynomial of degree k . For not necessarily regular graphs the above condition reads P k ( λ 0 ) > ‖ P k ‖ ∞ (‖ v ‖ 2 − 1) ⇒ D ⩽ k , where v is the positive eigenvector with smallest component equal to 1. To measure the accuracy of this result it seems interesting to investigate the graphs for which P k ( λ 0 ) = ‖ P k ‖ ∞ (‖ v ‖ 2 − 1), that we call boundary graphs. This has already been done for k = d − 1, and is undertaken in this paper for 1 ⩽ k d − 1. We present several families of such graphs, paying special attention to graphs with diameter D = k + 1.

From regular boundary graphs to antipodal distance-regular graphs

The problem was posed of determining the biclique partition number of the complement of a Hamiltonian path (Monson, Rees, and Pullman, Bull. Inst. Combinatorics and Appl. 14 (1995), 17–86). We define the complement of a path P, denoted

Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods

\overline{P}

The alternating polynomials and their relation with the spectra and conditional diameters of graphs

, as the complement of P in Km,n where P is a subgraph of Km,n for some m and n. We give an exact formula for the biclique partition number of the complement of a path. In particular, we solve the problem posed in [9]. We also summarize our more general results on biclique partitions of the complement of forests.

The vulnerability of the diameter of folded n -cubes

Abstract Recently, several results bounding above the diameter and/or the mean distance of a graph from its eigenvalues have been presented. They use the eigenvalues of either the adjacency or the Laplacian matrix of the graph. The main object of this paper is to compare both methods. As expected, they are equivalent for regular graphs. However, the situation is different for nonregular graphs: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance. This last statement follows from improved bounds on the mean distance obtained in the paper.

Graphs on alphabets as models for large interconnection networks

Abstract Given a graph Γ on n = ¦VΓ¦ vertices, the distance between two subgraphs Γ1, Γ2 ⊂ Γ, denoted by ∂(Γ1,Γ2), is the minimum among the distances between vertices of Γ1 and Γ2. For some integers 1 ⩽ s, t ⩽ n, the conditional (s, t)-diameter of Γ is then defined as D (s,t) = max Γ 1 , Γ 2 ⊂ Γ {∂(Γ 1 , Γ 2 ): ¦VΓ 1 ¦ = s, ¦VΓ 2 ¦ = t} . Let Γ have distinct eigenvalues λ > λ1 > λ2 > … > λd. For every k = 0, 1, …, d − 1, the k-alternating polynomial Pk is defined to be the polynomial of degree k and norm ‖P k ‖∞ = max 1⩽ / ⩽d {¦P k (λ l )¦} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by the authors to bound the (standard) diameter D ≡ D1,1 of Γ in terms of its eigenvalues. In this work we derive similar results for conditional diameters. For instance, it is shown that P k (λ)> ‖ν‖ 2 S −1 ⇒ D s,s ⩽k , where v is the (positive) eigenvector associated to λ, with minimum component 1. Similar results are given for locally regular digraphs by using the Laplacian spectrum. Some applications to the study of other parameters, such as the connectivity of Γ, are also discussed.

A UNIFIED APPROACH TO THE DESIGN AND CONTROL OF DYNAMIC MEMORY NETWORKS

Abstract Fault tolerance concern in the design of interconnection networks has arisen interest in the study of graphs such that the subgraphs obtained by deleting some vertices or edges have a moderate increment of the diameter. Besides the general problem, several particular families of graphs are worthy of consideration. Both the odd graphs and the n -cubes have been studied in this context. In this paper we deal with folded n -cubes, a much interesting family because: (i) like the n -cubes, their order is a power of 2, (ii) their diameter is half the diameter of the n -cube of the same order, while their degree only increases by one, and (iii) as we show, in a folded n -cube of degree Δ, the deletion of less than ⌊ 1 2 Δ⌋ − 1 vertices or edges does not increase the diameter of the graph, and the deletion of up to Δ − 1 vertices or edges increases it by at most one. This last property means that interconnection networks modelled by folded n -cubes are extremely robust.