Cai Heng Li

On isomorphisms of finite Cayley graphs: a survey

The isomorphism problem for Cayley graphs has been extensively investigated over the past 30 years. Recently, substantial progress has been made on the study of this problem, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained, and methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the finite simple group classification, through to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.

On Half-Transitive Metacirculant Graphs of Prime-Power Order

In this paper, infinitely many graphs of large valency which are half-transitive (that is vertex- and edge-transitive but not arc-transitive) are constructed, and a complete classification is given of half-transitive metacirculant graphs of order a p-power and valency less than 2p, where p is a prime. In particular, it is shown that, for any odd prime p, integers n?3 and k?2 such that k divides p?1, there are exactly (pn?2?1)/2+pn?3?1 nonisomorphic connected half-transitive metacirculants of order pn and valency 2k.

On edge-transitive Cayley graphs of valency four

A characterization is given of a class of edge-transitive Cayley graphs, providing methods for constructing Cayley graphs with certain symmetry properties. Various new half-arc transitive graphs are constructed.

On cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph Cay(G,S) is said to be normal if the group GR of right translations on G is a normal subgroup of the full automorphism group of Cay(G,S). In this paper, we prove that, for most finite simple groups G, connected cubic Cayley graphs of G are all normal. Then we apply this result to study a problem related to isomorphisms of Cayley graphs, and a problem regarding graphical regular representations of finite simple groups. The proof of the main result depends on the classification of finite simple groups.

On partitioning the orbitals of a transitive permutation group

Let G be a permutation group on a set Ω with a transitive normal subgroup M. Then G acts on the set Orbl(M, Ω) of nontrivial M-orbitals in the natural way, and here we are interested in the case where Orbl(M, Ω) has a partition P such that G acts transitively on P. The problem of characterising such tuples (M, G, Ω, P), called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where |P| is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where |P| = 2 exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the G-actions on Ω and on P, and gives some construction methods for TODs.

A class of finite symmetric graphs with 2-arc transitive quotients

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present paper studies the case where the size k of the blocks of D(B) satisfies k = v − 1. In the general case, where k = v − 1 > 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on B. We prove that D(B) contains no repeated blocks if and only if ΓB is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as ΓB for some graph Γ with these properties. We prove further that Γ[B,C]%Kv−1,v−1 if and only if ΓB is (G, 3)-arc transitive.

On isomorphisms of connected Cayley graphs

Abstract Let G be a finite group and Cay(G, S) the Cayley graph of G with respect to S. A subset S is called a CI-subset if, for any T ⊂ G, Cay(G, S) ≅ Cay(G, T) implies Sα = T for some α ∈ Aut(G). In this paper, we investigate the finite groups G in which every subset S with size at most m and (S⋋ = G is a CI-subset where m is a positive integer. As a corollary, we classify symmetric graphs of order p3 and of valency 2p where p is a prime.

On Finite s-Transitive Graphs of Odd Order

It is shown that, for a positive integer s, there exists an s-transitive graph of odd order if and only if s?3 and that, for s=2 or 3, an s-transitive graph of odd order is a normal cover of a graph for which there is an automorphism group that is almost simple and s-transitive.

On the Isomorphism Problem for Finite Cayley Graphs of Bounded Valency

For a subsetSof a groupGsuch that 1?SandS=S?1, the associated Cayley graph Cay(G,S) is the graph with vertex setGsuch that {x,y} is an edge if and only ifyx?1?S. Each ??Aut(G) induces an isomorphism from Cay(G,S) to the Cayley graph Cay(G,S?). For a positive integerm, the groupGis called anm-CI-group if, for all Cayley subsetsSof size at mostm, whenever Cay(G,S) ?Cay(G,T) there is an element ??Aut(G) such thatS?=T. It is shown that ifGis anm-CI-group for somem?4, thenG=U×V, where (|U|,|V|) =1,Uis abelian, andVbelongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification.

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where: (i) G has at most two fusion classes of order i for every i (23 examples); and (ii) any two elements of G of the same order are fused or inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.