A family of mixed graphs with large order and diameter 2

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.

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.

Discrete Mathematics, 2018

A mixed graph G can contain both (undirected) edges and arcs (directed edges). Here we derive an improved Moore-like bound for the maximum number of vertices of a mixed graph with diameter at least three. Moreover, a complete enumeration of all optimal (1, 1)-regular mixed graphs with diameter three is presented, so proving that, in general, the proposed bound cannot be improved.

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.

Discrete Mathematics, 2015

A directed strongly regular graph with parameters (n, k, t, λ, µ) is a k-regular directed graph with n vertices satisfying that the number of walks of length 2 from a vertex x to a vertex y is t if x = y, λ if there is an edge directed from x to y and µ otherwise. If λ = 0 and µ = 1 then we say that it is a mixed Moore graph. It is known that there are unique mixed Moore graphs with parameters (k 2 + k, k, 1, 0, 1), k ≥ 2, and (18, 4, 3, 0, 1). We construct a new mixed Moore graph with parameters (108, 10, 3, 0, 1) and also new directed strongly regular graphs with parameters (36, 10, 5, 2, 3) and (96, 13, 5, 0, 2). This new graph on 108 vertices can also be seen as an example of a so called multipartite Moore digraph. Finally we consider the possibility that mixed Moore graphs with other parameters could exist, in particular the first open case which is (40, 0,.

2016

The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, andŠiráň in [5], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-ˇ Siráň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.

Discrete Mathematics, 2008

The problem of determining the largest order n d,k of a graph of maximum degree at most d and diameter at most k is well known as the degree/diameter problem. It is known that n d,k M d,k where M d,k is the Moore bound. For d = 4, the current best upper bound for n 4,k is M 4,k − 1. In this paper we study properties of graphs of order M d,k − 2 and we give a new upper bound for n 4,k for k 3.

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.

1992

In this paper we give graphs with the largest known order for a given degree ∆ and diameter D. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the (∆, D) table for graphs.

Graphs and Combinatorics

The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while holding the other two fixed. Here we focus in bipartite Moore graphs, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with local bipartite Moore graphs. We find in some cases those local bipartite Moore graphs with local girths as close as possible to the local girths given by a bipartite Moore graph. Second, we construct a family of (q +2)-bipartite graphs of order 2(q 2 + q + 5) and diameter 3, for q a power of prime. These graphs attain the record value for q = 9 and improve the values for q = 11 and q = 13.