Xin Gui Fang

On cubic s -arc transitive Cayley graphs of finite simple groups

For a positive integer s, a graph Γ is called s-arc transitive if its full automorphism group AutΓ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S = S-1 and 1 ∉ S, let Γ = Cay(G, S) be the Cayley graph of G with respect to S and GR the set of right translations of G on G. Then GR forms a regular subgroup of AutΓ. A Cayley graph Γ = Cay(G, S) is called normal if GR is normal in AutΓ. In this paper we investigate connected cubic s-arc transitive Cayley graphs Γ of finite non-Abelian simple groups. Based on Lis work (Ph.D. Thesis (1996)), we prove that either Γ is normal with s ≤ 2 or G = A47 with s = 5 and AutΓ ≃ A48. Further, a connected 5-arc transitive cubic Cayley graph of A47 is constructed.

On the worst-case complexity of integer Gaussian elimination

Gaussian elimination is the basis for classical algorithms for computing canonical forms of integer matrices. Experimental results have shown that integer Gaussian elimination may lead to rapid growth of intermediate entries. On the other hand various polynomial time algorithms do exist for such computations, but these algorithms are relatively complicated to describe and understand. Gaussian elimination provides the simplest descriptions of algorithms for this purpose. These algorithms have a nice polynomial number of steps, but the steps deal with long operands. Here we show that there is an exponential length lower bound on the operands for a well-deflned variant of Gaussian elimination when applied to Smith and Hermite normal form calculation. We present explicit matrices for which this variant produces exponential length entries. Thus, Gaussian elimination has worst-case exponential space and time complexity for such applications. The analysis provides guidance as to how integer matrix algorithms based on Gaussian elimination may be further developed for better performance, which is important since many practical algorithms for computing canonical forms are so based.

5-Arc transitive cubic Cayley graphs on finite simple groups

In this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group A47; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two non-normal connected cubic arc-transitive Cayley graphs for finite nonabelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs.

ON THE AUTOMORPHISM GROUPS OF CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

On 1-Arc-regular Graphs

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G. Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain a sufficient condition under which one can guarantee that Cay(G, S) is 1-arc-regular. Finally, as an application of the result, we construct two infinite families of 1-arc-regular trivalent Cayley graphs with insoluble automorphism groups and, in particular, one of the families is not a Cayley graph.

On orbital regular graphs and Frobenius graphs

A graph is called a Frobenius graph if it is a connected orbital graph of a Frobenius group. In this paper, we show first that almost all orbital regular graphs are Frobenius graphs. Then we give a description of Frobenius graphs in terms of a family of (usually smaller) Frobenius graphs which are Cayley graphs for elementary abelian groups. Finally, based on this description, we obtain a formula for calculating the edge-forwarding index of Frobenius graphs.

A Family of Non-quasiprimitive Graphs Admitting a Quasiprimitive 2-arc Transitive Group Action

Let ? be a simple graph and let G be a group of automorphisms of ?. The graph is (G, 2)-arc transitive if G is transitive on the set of the 2-arcs of? . In this paper we construct a new family of (PSU(3, q2), 2)-arc transitive graphs ? of valency 9 such that Aut?=Z3.G , for some almost simple group G with socle PSU(3, q2). This gives a new infinite family of non-quasiprimitive almost simple graphs.

On the automorphism groups of symmetric graphs admitting an almost simple group

This paper investigates the automorphism group of a connected and undirected G-symmetric graph @C where G is an almost simple group with socle T. First we prove that, for an arbitrary subgroup M of [emailxa0protected] containing G, either T is normal in M or T is a subgroup of the alternating group Ak of degree k=|M@a:T@a|-|NM(T):T|. Then we describe the structure of the full automorphism group of G-locally primitive graphs of valency d, where [emailxa0protected]?20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group [emailxa0protected] for cubic symmetric graph @C admitting a finite almost simple group.