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A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p 2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given order. Based on the constructions of tetravalent half-arc-transitive graphs given by Marušič (J. Comb. Theory B 73:41-76, 1998), in this paper the connected tetravalent half-arc-transitive graphs of order 2pq are classified for distinct odd primes p and q.

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A graph Γ is said to be symmetric if its automorphism group Aut(Γ) is transitive on the arc set of Γ. Let G be a finite non-abelian simple group and let Γ be a connected pentavalent symmetric graph such that G ≤ Aut(Γ). In this paper, we show that if G is transitive on the vertex set of Γ, then either G Aut(Γ) or Aut(Γ) contains a non-abelian simple normal subgroup T such that G ≤ T and (G, T ) is one of 58 possible pairs of non-abelian simple groups. In particular, if G is arc-transitive, then (G, T ) is one of 17 possible pairs, and if G is regular on the vertex set of Γ, then (G, T ) is one of 13 possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, Ma and Wang in 2011.

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Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484-490] introduced the balanced hypercube BH n as an interconnection network topology for computing systems, and they proved that BH n is vertex-transitive. However, some other symmetric properties, say edge-transitivity and arctransitivity, of BH n remained unknown. In this paper, we solve this problem and prove that BH n is an arc-transitive Cayley graph. Using this, we also investigate some reliability measures, including super-connectivity, cyclic connectivity, etc., in BH n . First, we prove that every minimum edge-cut of BH n ðn ! 2Þ isolates a vertex, and every minimum vertex-cut of BH n ðn ! 3Þ isolates a vertex. This is stronger than that obtained by Wu and Huang which shows the connectivity and edge-connectivity of BH n are 2n. Second, Yang [Applied Mathematics and Computation 219 (2012) 970-975.] proved that for n ! 2, the super-connectivity of BH n is 4n À 4 and the super edge-connectivity of BH n is 4n À 2. In this paper, we proved that BH n ðn ! 2Þ is super-0 but not super-k 0 . That is, every minimum super edge-cut of BH n ðn ! 2Þ isolates an edge, but the minimum super vertex-cut of BH n ðn ! 2Þ does not isolate an edge. Third, we also obtain that for n ! 2, the cyclic connectivity of BH n is 4n À 4 and the cyclic edge-connectivity of BH n is 4ð2n À 2Þ. That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least 4n À 4 vertices (resp. 4ð4n À 2Þ edges) from BH n ðn ! 2Þ.

a b s t r a c tLet A n be the alternating group of degree n with n ≥ 3. SetThe alternating group graph, denoted by AG n , is defined as the Cayley graph on A n with respect to S. [J.-S. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315-326] introduced the alternating group graph AG n as an interconnection network topology for computing systems, and they proved that AG n is arc-transitive. In this work, it is shown that the full automorphism group of AG n is the semi-direct product R(A n ) Aut(A n , S), where R(A n ) is the right regular representation of A n and Aut(A n , S) = {α ∈ Aut(A n ) | S α = S} ∼ = S n−2 × S 2 . It follows from this result that AG n is arc-transitive but not 2-arc-transitive.

Abstract:A graph is vertex-transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1 / ∈ S and S = {s −1 | s ∈ S}. The Cayley graph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g ∈ G, s ∈ S}. Feng and Kwak [J Combin Theory B 97 (2007), 627-646; J Austral Math Soc 81 (2006), 153-164] classified all cubic symmetric graphs of order 4p or 2p 2 and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex-transitive nonCayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex-transitive graphs of order 2pq can be deduced. ᭧