Ernest Garriga

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.

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.

On the algebraic theory of pseudo-distance-regularity around a set☆

Let C be a connected graph with vertex set V, adjacency matrix A, positive eigenvector and corresponding eigenvalue 0. A natural generalization of distance-regularity around a vertex subset C V , which makes sense even with non-regular graphs, is studied. This new concept is called pseudo-distance-regularity, and its definition is based on giving to each vertex u 2 V a weight which equals the corresponding entry u of and “regularizes” the graph. This approach reveals a kind of central symmetry which, in fact, is an inherent property of all kinds of distance-regularity, because of the distance partition of V they come from. We come across such a concept via an orthogonal sequence of polynomials, constructed from the “local spectrum” of C, called the adjacency polynomials because their definition strongly relies on the adjacency matrix A. In particular, it is shown that C is “tight” (that is, the corresponding adjacency polynomials attain their maxima at 0) if and only if C is pseudo-distance-regular around C. As an application, some new spectral characterizations of distance-regularity around a set and completely regular codes are given.

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

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

\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.

Note: Number of walks and degree powers in a graph

Let Γ be a graph on n vertices, adjacency matrix A, and 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 ∥Pk∥∞ =max1⩽l⩽d {¦Pk(λl)¦} = 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 diameter D(Γ) of Γ in terms of its eigenvalues. Namely, it was shown that Pk(λ)> ∥v∥2 − 1 ⇒ D(Γ)⩽k, where v is the (positive) eigenvector associated to λ with minimum component 1. In this work we improve upon this result by assuming that some extra information about the structure of Γ is known. To this end, we introduce the so-called τ-adjacency polynomial Qτ. For each 0⩽τ⩽d, the polynomial Qτ is defined to be the polynomial of degree τ and norm ∥Qgt∥A = maxl⩽i⩽n{∥Qτ(A)ei∥} = 1 that attains maximum value at λ. Then it is shown that Pk(λ)>∥v∥2/Q2τ(λ) − 1 ⇒D(Γ)⩽k + 2τ. Some applications of the above results, together with new bounds for generalized diameters, are also presented.

Spectral and Geometric Properties of k-Walk-Regular Graphs

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.

Dual concepts of almost distance-regularity and the spectral excess theorem

This note deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the the number of all k-walks is upper bounded by the sum of the k-th powers of the degrees.