New mixed Moore graphs and directed strongly regular graphs

Ars Combinatoria, 2000

The nonexistence of digraphs with order equal to the Moore bound Md k = 1 + d+: : : +d k for d k > 1 has lead to the study of the problem of the existence of`almost' Moore digraphs, namely digraphs with order close to the Moore bound. In 1], it was shown that almost Moore digraphs of order Md k ; 1, degree d, diameter k (d k 3) contain either no cycle of length k or exactly one such cycle. In this paper we shall derive some further necessary conditions for the existence of almost Moore digraphs for degree and diameter greater than 1. As a consequence, for diameter 2 and degree d, 2 d 12,

Graphs and Combinatorics, 2019

The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree ≤ r and maximum directed out-degree ≤ z. Similarly one can search for the smallest possible k-geodetic mixed graphs with minimum undirected degree ≥ r and minimum directed out-degree ≥ z. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for k = 2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For k = 2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.

European Journal of Combinatorics, 2000

Considering a connected graph G with diameter D, we say that it is k-walk- regular, for a given integer k (0 ≤ k ≤ D), if the number of walks of length ℓ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed k. Thus, for k = 0, this definition

Journal of Combinatorial Theory, Series B, 2010

We define a class of digraphs involving differences in a group that generalizes Cayley digraphs, that we call difference digraphs. We define also a new combinatorial structure, called partial sum family, or PSF for short, from which we obtain difference digraphs that are directed strongly regular graphs. We give an infinite family of PSFs and we give also twelve sporadic ones that generate directed strongly regular graphs whose existence was previously undecided.

Journal of Combinatorial Theory, Series B, 2013

In this paper, we give a complete description of strongly regular graphs with parameters ((n 2 + 3n − 1) 2 , n 2 (n + 3), 1, n(n + 1)).

Journal of Interconnection Networks

A natural upper bound for the maximum number of vertices in a mixed graph with maximum undirected degree r, maximum directed out-degree z and diameter k is given by the mixed Moore bound. Graphs with order attaining the Moore bound are known as Moore graphs, and they are very rare. Besides, graphs with prescribed parameters and order one less than the corresponding Moore bound are known as almost Moore graphs. In this paper we prove that there is a unique mixed almost Moore graph of diameter k = 2 and parameters r = 2 and z = 1.

Cayley graphs are well known objects with interesting properties, also in the context of Moore graphs and digraphs. In 1978 Bos´ak extended Moore’s property to the mixed setting (with arcs allowed along with edges), in the so called mixed Moore graphs: those having a unique trail between pairs of vertices at a distance smaller than or equal to the diameter. In this paper we adapt Cayley’s construction to mixed graphs and we show certain mixed Moore graphs are Cayley while some other cannot be Cayley.

Linear Algebra and its Applications, 2010

In the degree-diameter problem, the only extremal graph the existence of which is still in doubt is the Moore graph of order 3250, degree 57 and diameter 2. It has been known that such a graph cannot be vertex-transitive. Also, certain restrictions on the structure of the automorphism group of such a graph have been known in the case when the order of the group is even. In this paper we further investigate symmetries and structural properties of the missing Moore (57, 2)-graph(s) with the help of a combination of spectral, group-theoretic, combinatorial, and computational methods. One of the consequences is that the order of the automorphism group of such a graph is at most 375.

1994

It is known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k ~ 3 there are no digraphs of order 'close' to, i.e., one less than, the Moore bound (14). For diameter 2, it is known that digraphs close to Moore bound exist for any degree because the line digraphs of complete digraphs are an example of such digraphs. However, it is not known whether these are the only digraphs close to Moore digraphs. In this paper, we shall consider the general case of directed graphs of diameter 2, degree d ~ 3 and with the number of vertices n = d + d 2 , that is, one less than the Moore bound. U sing the eigenvalues of the corresponding adjacency matrices we give a number of necessary conditions for the existence of such digraphs. Furthermore, for degree 3 we prove that there are no digraphs close to Moore bound other than the line digraph of K 4 • '