Joan Gimbert

Enumeration of almost Moore digraphs of diameter two

Abstract An almost Moore (d,2)-digraph is a regular directed graph of degree d>1, diameter k=2 and order n one less than the (unattainable) Moore bound. Their enumeration is equivalent to the characterization of binary matrices A fulfilling the equation I+A+A2=J+P, where J denotes the all-one matrix and P is a permutation matrix that commutes with A. In this paper we prove, using algebraic and graphical techniques, that if d>2 the previous equation has no solutions unless P=I. This allows us to complete the classification of the almost Moore (d,2)-digraphs up to isomorphisms. Thus, we conclude that there is only one (d,2)-digraph, namely the line digraph LKd+1 of the complete digraph Kd+1, apart from the particular case d=2 for which there are two more digraphs.

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these, digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.

Almost Moore digraphs are diregular

Abstract An almost Moore digraph is a digraph of diameter k⩾2 , maximum out-degree d⩾2 and order n=d+d 2 +⋯+d k , that is, one less than the Moore bound. It is easy to show that the out-degree of an almost Moore digraph is constant (=d) . In this note we prove that also the in-degree of an almost Moore digraph is constant (=d) , that is, every almost Moore digraph is diregular of degree d .

On the existence of ( d,k )-digraphs

Abstract A ( d , k )-digraph is a regular directed graph of degree d > 1, diameter k > 1, and order one less than the (unattained) Moore bound. If G is a ( d , k )- digraph, then for each vertex v ϵ V(G) there exists only one vertex, denoted by r ( v ) and called the repeat of v , such that there are exactly two v → r ( v ) walks of length less than or equal to k . The map r , which assigns to each vertex v ϵ V(G) the vertex r ( v ), is an automorphism of G , and its associated permutation matrix P satisfies the equation I + A + … + A k = J + P , where A is the adjacency matrix of G and J denotes the all-ones matrix. From this equation a close relationship between the spectrum of G and the cycle structure of the permutation r is derived. We use the characteristic polynomial of a ( d , k )-digraph to obtain some new necessary conditions for the cycle structure of the automorphism r of such a digraph. In particular, we apply these results to the study of the existence of the ( d , k )- digraphs in the cases k = 2,3. Finally, we prove that there is exactly one (4,2)-digraph, namely the line digraph of K 5 .

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.

The underlying line digraph structure of some (0,1)-matrix equations

From the theory ofHo0man polynomial, it is known that the adjacency matrix A ofa strongly connected regular digraph oforder n satis3es certain polynomial equation A l P(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coe5cients, and Jn is the n×n matrix of all ones. In this paper we present some su5cient conditions, in terms ofthe coe5cients of P(x), to ensure that all (0; 1)-matrices satisfying such an equation with l? 0 have an underlying line digraph structure, that is to say, for any solution A there exists a (0; 1)-matrix C satisfying P(C)= Jn=dl and the associated (d-regular) digraph of A, � (A), is the lth iterated line digraph of � (C). As a result, we simplify the study of some digraph classes with order functions asymptotically attaining the Moore bound. ? 2002 Elsevier Science B.V. All rights reserved.

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

Radial Moore graphs of radius three

The maximum number of vertices in a graph of specified degree and diameter cannot exceed the Moore bound. Graphs achieving this bound are called Moore graphs. Because Moore graphs are so rare, researchers have considered various relaxations of the Moore graph constraints. Since the diameter of a Moore graph is equal to its radius, one can consider graphs in which the condition on the diameter is relaxed, by one, while the condition on the radius is maintained. Such graphs are called radial Moore graphs. It has previously been shown that radial Moore graphs exist for all degrees when the radius is two. In this paper, we extend this result to radius three. We also construct examples that settle the existence question for a few new cases, and summarize the state of knowledge on the problem.

Ranking measures for radially Moore graphs

For graphs with maximum degree d and diameter k, an upper bound on the number of vertices in the graphs is provided by the well‐known Moore bound (denoted by Md,k). Graphs that achieve this bound (Moore graphs) are very rare, and determining how close one can come to the Moore bound has been a major topic in graph theory. Of particular note in this regard are the cage problem and the degree/diameter problem. In this article, we take a different approach and consider questions that arise when we fix the number of vertices in the graph at the Moore bound, but relax, by one, the diameter constraint on a subset of the vertices. In this context, regular graphs of degree d, radius k, diameter k + 1, and order equal to Md,k are called radially Moore graphs. We consider two specific questions. First, we consider the existence question (extending the work of Knor), and second, we consider some natural measures of how well a radially Moore graph approximates a Moore graph.

On the Weak Distance-regularity of Moore-type Digraphs

We prove that Moore digraphs, and some other classes of extremal digraphs, are weakly distance-regular in the sense that there is an invariance of the number of walks between vertices at a given distance. As weakly distance-regular digraphs, we then compute their complete spectrum from a ‘small’ intersection matrix. This is a very useful tool for deriving some results about their existence and/or their structural properties. For instance, we present here an alternative and unified proof of the existence results on Moore digraphs, Moore bipartite digraphs and, more generally, Moore generalized p-cycles. In addition, we show that the line digraph structure appears as a characteristic property of any Moore generalized p-cycle of diameter D ≥ 2p.