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A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, a complete classification is given of tetravalent vertex-transitive non-Cayley graphs of order $$2p^2$$ for any prime p. [ABSTRACT FROM AUTHOR]

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p² is given. [ABSTRACT FROM AUTHOR]

A finite graph is one-regular if its automorphism group acts regularly on the set of its arcs. In the present paper, tetravalent one-regular graphs of order 7p2, where p is a prime, are classified using computer algebra tools. [ABSTRACT FROM AUTHOR]

Abstract: A graph is symmetric if its automorphism group is transitive on the arc set of the graph. In this paper, we classify connected cubic symmetric graphs of order for each prime . All those symmetric graphs are explicitly constructed as normal Cayley graphs on some groups of order , and their automorphism groups are determined. There is a unique connected cubic symmetric graph of order . All connected cubic symmetric graphs of order for are regular covers of the three dimensional hypercube , and consist of four infinite families, of which two families exist if and only if and the other two families exist for each odd prime . In each family, there is a unique graph for a given order. [Copyright &y& Elsevier]

A Cayley graph (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to for n=2, and to for n > 2. Furthermore, we determine all pairs of G and S such that Γn=(G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p. [ABSTRACT FROM AUTHOR]

Let s be a positive integer. A graph is s -transitive if its automorphism group is transitive on s-arcs but not on ( s + 1)-arcs. Let p be a prime. Zhou (Discrete Math 309:6081-6086, ) classified tetravalent s-transitive graphs of order 4 p. In this article a complete classification of tetravalent s-transitive graphs of order 4 p is given. [ABSTRACT FROM AUTHOR]

Abstract: In this article a complete classification of tetravalent arc-transitive graphs of order twice a product of two primes is given. [Copyright &y& Elsevier]

Abstract: A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215–232], there is no semisymmetric graph of order or for a prime and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order , Discrete Math. 274 (2004) 187–198], there exists a unique cubic semisymmetric graph of order , the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order exists if and only if is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly. [Copyright &y& Elsevier]