Yan-Quan Feng

Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

Let T be a set of transpositions of the symmetric group Sn. The transposition graph Tra(T) of T is the graph with vertex set {1, 2, ..., n} and edge set {ij|(i j) ∈ T}. In this paper it is shown that if n ≥ 3, then the automorphism group of the transposition graph Tra(T) is isomorphic to Aut(Sn, T) = {α ∈ Aut(Sn)|Tα = T} and if T is a minimal generating set of Sn then the automorphism group of the Cayley graph Cay(Sn, T) is the semiproduct R(Sn) Aut(Sn, T), where R(Sn) is the right regular representation of Sn. As a result, we generalize a theorem of Godsil and Royle [C.D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001, p. 53] regarding the automorphism groups of Cayley graphs on Sn : if T is a minimal generating set of Sn and the automorphism group of the transposition graph Tra(T) is trivial then the automorphism group of the Cayley graph Cay(Sn, T) is isomorphic to Sn.

One-regular cubic graphs of order a small number times a prime or a prime square

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2 p or 2 p 2 where p is a prime if and only if 3 is a divisor of p 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4 p or 4 p 2 .

Tetravalent half-arc-transitive graphs of order p4

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. It is known that for a prime p there is no tetravalent half-arc-transitive graphs of order p or p^2. Xu [M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified the tetravalent half-arc-transitive graphs of order p^3. As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p^4. It shows that there are exactly p-1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p^4 for each odd prime p.

Conditional Diagnosability of Alternating Group Graphs

Let An be the alternating group of degree n with n ≥ 3. Set S = {(1 2 i), (1 i 2)| 3 ≤ i ≤ n}. The alternating group graph, denoted by AGn, is defined as the Cayley graph on An with respect to S. Jwo et al. [Networks 23 (1993) 315-326] introduced alternating group graph AGn as an interconnection network topology for computing systems. Conditional diagnosability, a new measure of diagnosability introduced by Lai et al. [IEEE Transactions on Computers 54(2) (2005) 165-175] can better measure the diagnosability of regular interconnection networks. This paper determines that under PMC-model the conditional diagnosability of AGn is 4 for n = 4 and 6n -18 for each n ≥ 5.

Tetravalent half-transitive graphs of order 4p

A graph is half-transitive if its automorphism group acts transitively on its vertex set and edge set, but not on its arc set. In this paper, the tetravalent half-transitive graphs of order 4p are classified for each prime p. It is shown that there are no tetravalent half-transitive Cayley graphs of order 4p and a tetravalent half-transitive non-Cayley graph of order 4p exists if and only if p-1 is divisible by 8, which is unique for a given order.

Two node-disjoint paths in balanced hypercubes

The balanced hypercube BHn proposed by Wu and Huang is a variation of the hypercube. It has been proved that the balanced hypercube is a node-transitive and bipartite graph. Assume that the nodes are divided into two bipartite node sets X and Y,u and x are two different nodes in X, and v and y are two different nodes in Y. In this paper, we prove that there exist two node-disjoint paths P[x,y] and R[u,v] in BHn, and V(P[x,y])∪V(R[u,v])=V(BHn), where n⩾1. The Hamiltonian laceability of BHn which was proved by Xu et al. is also obtained from the corollary of our result.

Automorphism groups of Cayley digraphs

Let G be a group and S ⊂ G with 1 � S. A Cayley digraph Cay(G, S) on G with respect to S is the digraph with vertex set G such that, for x, y ∈ G , there is a directed edge from x to y whenever yx −1 ∈ S.I fS −1 = S, then Cay(G, S) can be viewed as an (undirected) graph by identifying two directed edges (x, y) and ( y, x) with one edge {x, y}. Let X = Cay(G, S) be a Cayley digraph. Then every element g ∈ G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. The Cayley digraph Cay(G, S) is said to be normal if R(G) ={ R(g)|g ∈ G} is a normal subgroup of the automorphism group of X. In this paper we shall give a brief survey of recent results on automorphism groups of Cayley digraphs concentrating on the normality of Cayley digraphs. Throughout this paper graphs or digraphs (directed graphs) are finite and simple unless specified otherwise. For a (di)graph X , we denote by V (X ), E(X ) and Aut(X ) the vertex set, the edge set and the automorphism group of X , respectively. A (di)graph is said to be vertex-transitive or edge- transitive if Aut(X ) acts transitively on V (X ) or E(X ), respectively. Note that for an (undirected) graph X , each edge {u, v} of X gives two ordered pairs (u, v) and (v, u), called arcs of X. Thus we sometimes, if necessary, view a graph X as a digraph. Let G be a group and S a subset of G such that 1 �∈ S. The Cayley digraph Cay(G, S) on G with respect to S is defined as the directed graph with vertex set G and edge set {(g, sg) | g ∈ G, s ∈ S}. For a Cayley digraph X = Cay(G, S), we always call | S | the valency of X for convenience. If S is symmetric, that is, if S −1 ={ s −1 | s ∈ S} is equal to S, then Cay(G, S) can be viewed as an undirected graph by identifying two oppositely directed edges with one undirected edge. We sometimes call a Cayley digraph Cay(G, S) a Cayley graph if S is symmetric, and say Cay(G, S) a directed Cayley graph to emphasize S −1 � S. Let X = Cay(G, S) be a Cayley digraph. Consider the action of G on V (X ) by right multiplica- tion. Then every element g ∈ G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. Set R(G) ={ R( g) | g ∈ G}. Then R(G) is a subgroup of Aut(X ) and R(G) ∼ G. Thus X is a vertex-transitive digraph. Clearly, R(G) acts regularly on vertices, that is, R(G) is transitive on vertices and only the identity element of R(G) fixes any given vertex. Further, it is well-known that a digraph Y is isomorphic to a Cayley digraph on some group G if and only if its automorphism group contains a subgroup isomorphic to G, acting regularly on the vertices of Y (see (5, Lemma 16.3)). Noting that R(G) is regular on V (X ), it implies Aut(X ) = R(G)Aut(X )1.

Hamiltonian cycle embedding for fault tolerance in balanced hypercubes

Abstract The balanced hypercube BH n , defined by Wu and Huang, is a variant of the hypercube network. Yang proposed that fault tolerance of balanced hypercube BH n is an important issue in parallel computing which needs further study (Yang, 2010) [24]. In this paper, we prove that there exists a fault-free Hamiltonian path between any two adjacent vertices in BH n with 2 n - 2 faulty edges. As a corollary, we derive that for any fault-free edge e, there exists a fault-free Hamiltonian cycle containing e in BH n with 2 n - 2 faulty edges which is optimal in the sense of the number of faulty edges.

Constructing an Infinite Family of Cubic 1-Regular Graphs

A graph is 1- regular if its automorphism group acts regularly on the set of its arcs. Miller J. Comb. Theory, B , 10 (1971), 163?182] constructed an infinite family of cubic 1-regular graphs of order 2 p, where p? 13 is a prime congruent to 1 modulo 3. Maru?i? and Xu J. Graph Theory, 25 (1997), 133?138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al.J. Aust. Math. Soc. A, 56 (1994), 391?402] constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller?s construction can be generalized to an infinite family of cubic 1-regular graphs of order 2n , where n? 13 is odd such that 3 divides ?(n), the Euler function of n. In this paper, we construct an infinite family of cubic 1-regular graphs with order 8(k2+k+ 1)(k? 2) as cyclic-coverings of the three-dimensional Hypercube.

Hexavalent half-arc-transitive graphs of order 4p

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. It was shown by [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent half-transitive graphs of order 4p, European J. Combin. 28 (2007) 726-733] that all tetravalent half-arc-transitive graphs of order 4p for a prime p are non-Cayley and such graphs exist if and only if p-1 is divisible by 8. In this paper, it is proved that each hexavalent half-arc-transitive graph of order 4p is a Cayley graph and such a graph exists if and only if p-1 is divisible by 12, which is unique for a given order. This result contributes to the classification of half-arc-transitive graphs of order 4p of general valencies.