In this paper, we introduce a class of double coset cayley digraphs induced by right solvable ward groupoids. These class of graphs can be viewed as a generalization of double coset cayley graphs induced by groups. Further, many graph properties are expressed in terms of algebraic properties.

We construct a polynomial-time algorithm that given a graph X with 4p vertices (p is prime), finds (if any) a Cayley representation of X over the group C2 × C2 × Cp. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order 4p can be done in polynomial time.

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.

For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3)k. By comparison with other available results in this area we show that, for all sufficiently large d and k, our family gives the current largest known Cayley graphs of degree d and diameter k.

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

Self-complementary Cayley graphs are useful in the study of Ramsey numbers, but they are relatively very rare and hard to construct. In this paper, we construct several families of new self-complementary Cayley graphs of order p4 where p is a prime and congruent to 1 modulo 8.

New criteria for which Cayley graphs of cyclic groups of any order can be completely determined–up to isomorphism–by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley graphs of cyclic groups with the same list of eigenvalues of their adjacency matrices will be presented.

Metacirculants are a basic and well-studied family of vertex-transitive graphs, and weak metacirculants are generalizations of them. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. This paper is devoted to the study of weak metacirculants with odd prime power order. We first prove that a weak metacirculant of odd prime power order is a metacir...

Arising from complete Cayley graphs Γn of odd cyclic groups Zn, an asymptotic approach is presented on connected labeled graphs whose vertices are labeled via equallymulticolored copies of K4 in Γn with adjacency of any two such vertices whenever they are represented by copies of K4 in Γn sharing two equally-multicolored triangles. In fact, these connected labeled graphs are shown to form a fam...