We prove that the automorphism group of a self-complementary metacirculant is either soluble or has A 5 as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.

In this short paper, we characterise graphs of order pq with p, q prime which are self-complementary and vertex-transitive.2010 Mathematics subject classification: primary 05C25; secondary 05E18.

Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

The aim of this paper is to prove, mainly, a common fixed point theorem for six self mappings and its consequences under the condition of weakly compatible mappings in a Quasi-Gauge space.

The notion of rational F-contractions using α -admissibility of type-S in b-metric-like spaces is introduced and the new fixed and periodic point theorems are proved for such mappings. Numerical examples are illustrated to check the efficiency and applicability of our fresh findings. It is also observed that some of the works reported in the literature are the particular cases of the present study.