In this paper we introduce the Cayley digraph structure. This can be considered as a generalization of Cayley digraph. We prove that all Cayley digraph structures are vertex transitive. Many graph theoretic properties are studied in terms of algebraic properties.

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G . Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain...

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

In this paper, we present a new extension of the butterfly digraph, which is known as one of the topologies used for interconnection networks. The butterfly digraph was previously generalized from binary to d-ary. We define a new digraph by adding a signed label to each vertex of the d-ary butterfly digraph. We call this digraph the dihedral butterfly digraph and study its properties. Furthermo...

1. Dejter, I., Giudici, R.E.: On Unitary Cayley graphs, JCMCC, 18,121-124. 2. Brrizbitia, P and Giudici, R.E. 1996, Counting pure k-cycles in sequence of Cayley graphs, Discrete math., 149, 11-18. 3. Madhavi, L and Maheswari, B.2009, Enumeration of Triangles and Hamilton Cycles in Quadratic residue Cayley graphas, Chamchuri Journal of Mathematics, 1,95-103. 4. Madhavi, L and Maheswari, B. 2010,...

Let us say that a Cayley graph of a group G of order n is a Černý Cayley graph if every synchronizing automaton containing as a subgraph with the same vertex set admits a synchronizing word of length at most (n− 1)2. In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of Černý Cayley graphs.

Given an integer n, one defines the unitary Cayley graph, denoted Cay(Zn,Zn), to be the graph whose vertex set is Zn, the integers modulo n, with an edge between two vertices x, y if x− y is a unit in (the ring) Zn. Unitary Cayley graphs have been studied as objects of independent interest (see, for example, [3], [2], [7], [8], [9]) but are of particular relevance in the study of graph represen...

Graph Theory has been realized as one of the most useful branches of Mathematics of recent origin, finding widest applications in all most all branches of sciences, social sciences, and engineering and computer science. Nathanson[8] was the pioneer in introducing the concepts of NumberTheory, particularly, the “Theory of congruences” i n Graph Theory, thus paving way for the emergence of a new ...

The point we try to get across is that the generalization of the counterparts of the matroid theory in Cayley graphs since the matroid theory frequently simplify the graphs and so Cayley graphs. We will show that, for a Cayley graph ΓG, the cutset matroid M ∗(ΓG) is the dual of the circuit matroid M(ΓG). We will also deduce that if Γ ∗ G is an abstract-dual of a Cayley graph Γ, then M(Γ∗ G ) is...