This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every subgroup G automorphism group a nontrivial design primitive affine or almost simple type. Moreover, we classify designs admitting flag transitive with socle PSL(n,q) for some n≥3. Alongside this analysis give construction 2-(v,k−1,k−2) given 2-(...

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. Cheng and Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42(1987) 196-211] proved that there is no half-arc-transitive graph of order 2p, and Alspach and Xu [ 12 -transitive graphs of order 3p, J. Algebraic Combin. 3(1994) 347-355] c...

A graph G admits an H-covering if every edge of belongs to a subgraph isomorphic given H. is said be H-magic there exists bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) constant, for H′ In particular, H-supermagic f(V(G))={1,2,…,|V(G)|}. When H complete K2, H-(super)magic labeling edge-(super)magic labeling. Suppose F-covering and two graphs F We define (...

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary gr...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...

In this paper highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation α on N there is another permutation β on N such that the subgroup generated by α and β is highly transitive. The Baire category method is u...

All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points. This note contains a proof of the following result. Theorem. Let D be a symmetric design with v > 2k such that Aut D is 2-transitive on points. Then D is one o f the following: (i) a projective space; (ii) the unique Hadamard design with v = I 1 and k = 5; (iii) a unique design with v = 1...

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups Zn, symmetric groups Sn and quantum symmetric groups Qn, by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.