The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Algebraic Graph Theory: The multiplicative expansion

Cubic vertex-transitive graphs on up to 1280 vertices

A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

Classical Groups, Derangements and Primes

A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.

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A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

Let A be an abelian group and let ź be the automorphism of A defined by: ź: a ź aź1. A Cayley graph Γ = Cay(A,S) is said to have an automorphism group as small as possible if Aut(Γ)=Aź . In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.

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Cayley graphs on abelian groups

The main result of this paper is that, if $\Gamma$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $\Gamma$, then either $\Gamma$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or $2|G_v|\log_2(|G_v|/2)\leq |\V\Gamma|$ and that this last bound is tight. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs.

Gabriel Verret

Papers

Cubic vertex-transitive graphs on up to 1280 vertices

A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

Cayley graphs on abelian groups

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs