Let Γ be an abelian group and B be a subset of Γ. The addition Cayley graph G′ = Cay(Γ, B) is the graph having the vertex set V (G′) = Γ and the edge set E(G′) = {ab : a + b ∈ B}, where a, b ∈ Γ. For a positive integer n > 1, the unitary addition Cayley graph Gn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b a...

Cycle is one of the most fundamental graph classes. For a given graph, it is interesting to find cycles of various lengths as subgraphs in the graph. The Cayley graph Cay(Sn, S) on the symmetric group has an important role for the study of Cayley graphs as interconnection networks. In this paper, we show that the Cayley graph generated by a transposition set is vertex-bipancyclic if and only if...

Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the˜∂-equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner, Leray, Koppelman type used in complex analysis here are proved in the new generalized form for f...

for two normal edge-transitive cayley graphs on groups h and k which have no common direct factor and gcd(jh=h ′j; jz(k)j) = 1 = gcd(jk=k ′j; jz(h)j), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.

For a group T and a subset S of T , the bi-Cayley graph BCay(T, S) of T with respect to S is the bipartite graph with vertex set T×{0, 1} and edge set {{(g, 0), (sg, 1)} | g ∈ T, s ∈ S}. In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infin...

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an A...

An automorphism α of a Cayley graph Cay(G,S) of a group G with connection set S is color-preserving if α(g, gs) = (h, hs) or (h, hs−1) for every edge (g, gs) ∈ E(Cay(G,S)). If every color-preserving automorphism of Cay(G,S) is also affine, then Cay(G,S) is a CCA (Cayley color automorphism) graph. If every Cayley graph Cay(G,S) is a CCA graph, then G is a CCA group. Hujdurović, Kutnar, D.W. Morr...

We show that almost every Cayley graph Γ of an abelian group G of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph Γ of an abelian group G of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of G (that is, GL/Aut(Γ)).

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.