M.A. Fiol

The connectivity of large digraphs and graphs

Abstract Let C=( V, A) be a digraph on n vertices with maximum degree A and diameter D, so that n h{n(A,D-l)+n(A,1-1)-2}+AD-‘+‘+l; G is super-l if n>G{n(A,D-l-l)+n(A,I-l)}+Av~‘, where I stands for a parameter related with the number of short paths. Similar results are given for graphs (in this case it turns out that I=L(g- 1)/2 J where g stands for the girth). 1. Notation and basic results Let G = (V, A) denote a digraph with the (finite) set of uertices V= V(G) and the set of arcs A = A(G),

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.

On pseudo-distance-regularity

Abstract The concept of (local) pseudo-distance-regularity, recently introduced, is a natural generalization of distance-regularity, intended for not necessarily regular graphs. We study here some properties of locally pseudo-distance-regular graphs and give some new characterization of such structures. As a consequence, some new characterizations of distance-regular graphs are also derived.

Graphs on alphabets as models for large interconnection networks

Abstract Graphs on alphabets are constructed by labelling the vertices with words on a given alphabet, and specifying a rule that relates pairs of different words to define the edges. They have proved to be quite suitable to model large interconnection networks since their structure usually provides efficient routing algorithms. The aim of this paper is to present several infinite families of such graphs with a large number of vertices for given values of their diameter and maximum degree.

Some spectral and quasi-spectral characterizations of distance-regular graphs

In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.

Algebraic characterizations of regularity properties in bipartite graphs

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph ? is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the Hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.

The Spectra of Wrapped Butterfly Digraphs

The knowledge of the spectrum of a (di)graph is relevant for estimating some of its structural properties, which provide information on the topological and communication properties of the corresponding networks. Among these properties, we have, for instance, edge-expansion and node-expansion, bisection width, diameter, maximum cut, connectivity, and partitions. In this paper, we determine the complete spectra (eigenvalues and multiplicities) of wrapped butterfly digraphs.

On the hierarchical product of graphs and the generalized binomial tree

In this article we follow the study of the hierarchical product of graphs, an operation recently introduced in the context of networks. A well-known example of such a product is the binomial tree which is the (hierarchical) power of the complete graph on two vertices. An appealing property of this structure is that all the eigenvalues are distinct. Here we show how to obtain a graph with this property by applying the hierarchical product. In particular, we propose a generalization of the binomial tree and study some of its main properties.

On Moore bipartite digraphs: ON MOORE BIPARTITE DIGRAPHS

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d1; d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 k 4. This paper deals with

Periodic Tilings as a Dissection Method

1. INTRODUCTION. In geometry it is sometimes useful to prove results about the equivalence of two areas or volumes, here referred to as (equivalent) tiles, by using the dissection method. This method consists of cutting both tiles into the same (finite) number of pieces, or subtiles, in such a way that each piece of a tile has a corresponding congruent piece from the other tile. In this case, the tiles are said to be congruent by addition or (geometrically) equidecomposable. In fact, according to the classic theorem of Bolyai-Gerwin, any two polygonal tiles with equal area are congruent by addition, and the dissection can always be done with straightedge and compasses [1]. However, as Dehn [3] showed, the analogous result for polyhedra does not hold, since a regular tetrahedron and a cube of the same volume are not equidecomposable. For two comprehensive accounts of interesting dissections, see [15] and [7]. Another related approach is to prove that the equivalent tiles are congruent by subtraction or equicomplementable. This means that some corresponding pairs of congruent pieces can be added on in order to get two new tiles that are congruent by addition. In this context, Sydler [20] proved that two equivalent polygons or polyhedra are equicomplementable if and only if they are equidecomposable. Sometimes we require not only that the equivalent tiles be equidecomposable (or equicomplementable) but also that they can be transformed into one another by using certain types of rigid motions (or isometries), such as translations and rotations. Thus, since the allowed transformations always form a group, say G, we then speak about G-equidecomposability. Another classical result in this context is the Hadwiger-Glur theorem [14]: two polygonal planar regions with equal area are always Gs-equidecomposable, where Gs is the group of translations and central symmetries. In fact, Gs is the minimum subgroup of the whole group GK of isometries of the plane bearing this property. Given a discrete group G of transformations of Euclidean n-space, we say that a tile G-periodically tessellates (or, simply, G-tessellates) the space when the transformations of the tile by the elements of G completely cover the space without gaps or overlaps. For example, if G is generated by some translations, the entire space is filled with proper translations (neither rotations nor symmetries are allowed) of the tile, called translates. For any periodic tessellation there is a corresponding underlying lattice, or set of points representing the feasible translations. The reader can find a very extensive treatment of periodic (and nonperiodic) tessellations in [11]. Our aim here is to show the close relationship between some congruent (by addition or subtraction) tiles and their corresponding congruent (that is, with the same lattice) periodic tilings. In particular, this provides a method of obtaining (old and new) dissection proofs of geometric results. The basic concepts and results concerning this approach are dealt with in Section 2, whereas Sections 3 and 4 are devoted to giving some examples of applications of the method.