Nonexistence of an extremal graph of a certain type (original) (raw)

Radial Moore graphs of radius three

Discrete Applied Mathematics, 2012

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.

Graphs of order two less than the Moore bound

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.

Search for properties of the missing Moore graph

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.

On mixed Moore graphs

Discrete Mathematics, 2007

The Moore bound for a directed graph of maximum out-degree d and diameter k is M d,k = 1 + d + d 2 + · · · + d k . It is known that digraphs of order M d,k (Moore digraphs) do not exist for d > 1 and k > 1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is

Nonexistence of almost Moore digraphs of diameter three

The Electronic Journal of Combinatorics, 2008

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound M (d, k) = 1 + d + · · · + d k , where d > 1 and k > 1 denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when d = 2, 3 or k = 2. In this paper, we prove that almost Moore digraphs of diameter k = 3 do not exist for any degree d.

The Unique Mixed Almost Moore Graph with Parameters k = 2, r = 2 and z = 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.

Complete characterization of almost Moore digraphs of degree three

Journal of Graph Theory, 2005

It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 112–126, 2005

The Unique Mixed Almost Moore Graph with Parameters k = 2, r = 2 and z = 1

Journal of Interconnection Networks, 2017

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. In this paper we prove that there is a unique mixed graph of diameter k = 2 and parameters r = 2 and z = 1 containing the largest possible number of vertices, which in this case is one less than the corresponding mixed Moore bound. Mixed graphs with prescribed parameters and order one less than the corresponding Moore bound are known as mixed almost Moore graphs.

On New Record Graphs Close to Bipartite Moore 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.

Further results on almost Moore digraphs

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,