The complementary prism of a graph Γ is the ΓΓ̄, which formed from union and its complement Γ̄ by adding an edge between each pair identical vertices in Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive prisms. It was recently proved author that ΓΓ̄ core, i.e. all endomorphisms are automorphisms, whenever vertex-transitive, self-complementary, either core or complete graph. ...

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.

In this paper, a complete classification of finite simple cubic vertex-transitive graphs girth 6 is obtained. It proved that every such graph, with the exception Desargues graph on 20 vertices, either skeleton hexagonal tiling torus, truncation an arc-transitive triangulation closed hyperbolic surface, or 6-regular respect to dihedral scheme. Cubic larger than are also discussed.

In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive gr...