A Cayley graph Cay(G,S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G,S). In this paper, all connected tetravalent non-normal Cayley graphs of order 4p are constructed explicitly for each prime p. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of orde...

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and corefree), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various c...

‎‎Set X = { M11‎, ‎M12‎, ‎M22‎, ‎M23‎, ‎M24‎, ‎Zn‎, ‎T4n‎, ‎SD8n‎, ‎Sz(q)‎, ‎G2(q)‎, ‎V8n}‎, where M11‎, M12‎, M22‎, ‎M23‎, ‎M24 are Mathieu groups and Zn‎, T4n‎, SD8n‎, ‎Sz(q)‎, G2(q) and V8n denote the cyclic‎, ‎dicyclic‎, ‎semi-dihedral‎, ‎Suzuki‎, ‎Ree and a group of order 8n presented by                                      V8n = < a‎, ‎b | a^{2n} = b^{4} = e‎, ‎ aba = b^{-1}‎, ‎ab^{...

The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as cage problem. This problem natural extension to hypergraphs, where we may ask number vertices in $d$-regular, $r$-uniform hypergraph given (Berge) $g$. We show that these two problems are fact very closely linked. By extending ideas Cayley graphs context, find hypergraphs various para...

A Cayley graph Γ = Cay(G,S) is said to be normal for a finite group G, if the right regular representation R(G) is normal in the full automorphism group Aut(Γ) of Γ. In this paper we investigate the normality of Cayley graphs of groups of order a product of two distinct primes, by determining all nonnormal Cayley graphs of these groups.

A Cayley graph Cay(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 Cay(G,S). For a positive integer n, let Γn be a graph having 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 ...

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t-Cayley and t-Gayley polytopes), and to the most general two-v...