Meng Jixiang

THE DIAMETERS OF ALMOST ALL CAYLEY DIGRAPHS

LetG be a finite group of ordern andS be a subset ofG not containing the identity element ofG. Letp (0<p<1) be a fixed number. We define the set of all labelled Cayley digraphsX(G,S) (S<-G\{1}) ofG as a sample space and assign a probability measure by requiringP(aεS)=p for anya∈G\{1}. Here it is shown that the probability of the set of Cayley digraphs ofG with diameter 2 approaches 1 as the ordern ofG approaches infinity.

Almost all Cayley graphs are hamiltonian

It has been conjectured that there is a hamiltonian cycle in every finite connected Cayley graph. In spite of the difficulty in proving this conjecture, we show that almost all Cayley graphs are hamiltonian. That is, as the order n of a groupG approaches infinity, the ratio of the number of hamiltonian Cayley graphs ofG to the total number of Cayley graphs ofG approaches 1.

Isomorphisms of circulant diagaphs

LetS⊂Zn−{0}. The circulant digraphDCn(S) is a directed graph with vertex setZn and arc set {(i,i+s): i ∈ Zn, s ∈ S}. A. Adam conjectured thatDCn(S)≅DCn(T) if and only ifT=uS for some unitu modn. In this paper we prove that the conjecture is true ifS is a minimal generating set ofZn and thus determine the full automorphism groups of such digraphs. The methods we employ are new and easy to be understood.

Onnectivities of minimal cayley coset digraphs

We prove that the connectivities of minimal Cayley coset digraphs are their regular degrees.

On circulant digraphs with regular automorphism groups

We prove that the cyclic groupZ n (n ≥ 3) has ak-regular digraph regular representation if and only if 0We prove that the cyclic groupZn (n ≥ 3) has ak-regular digraph regular representation if and only if 0

Connectivities of random circulant digraphs

AbstractIn this paper, we prove that almost all circulant digraphs are strongly connected. Furthermore, for any given positive integerm, we show that almost every circulant digraphC has connectivity at least

Superconnected and Hyperconnected 6-Regular Transitive Graphs

\frac{m}{{1 + m}} + \left( {d\left( C \right) + 1} \right)

Path Extendability of s-vertex Connected Graphs

, whered(C) is the vertex degree ofC.