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We investigate some properties of Cayley fuzzy graphs on groups in terms of algebraic structures. And we also discuss the connectedness on it. In this paper, we prove that the Cayley fuzzy graph induced by (Zm × Zn,+, ν) is the disjoint union of Cay(aH, R) provided ν(h) > 0 h H and (h0) = 0 - h0 - Zm ×Zn\H, where H is the cyclic subgroup of order n in Zm × Zn. [ABSTRACT FROM AUTHOR]

In 1878, Cayley introduced graphs of groups. Later, graphs of semigroups were introduced as generalization of Cayley graphs of groups. In 1940, Rees introduced matrix semigroups. In this paper, we describe some properties of Cayley graphs of Rees matrix semigroups. We see that, for a group G and for any arbitrary non-empty finite sets I and Λ, the Cayley graph of the Rees matrix semigroup S = M(G; I,Λ; P) relative to any Green’s equivalence L-class of S has a Hamiltonian decomposition consisting of |Λ| components.Moreover, when |G| × |I| is odd, we show that the decomposition is Eulerian. [ABSTRACT FROM AUTHOR]

The most natural source of problems in semigroup theory is the study of semigroup of transformations. It is well known that Green's relations play an important role in the theory of semigroups. The definition of Cayley graphs of groups was introduced by Arthur Cayley in 1878 and Cayley graphs of semigroups are general- izations of Cayley graphs of groups. In this paper, we study the Cayley graphs of Symmetric inverse semigroup relative to Green's equivalence R-Class. [ABSTRACT FROM AUTHOR]

Let G be the dihedral group Dn and Cay(G, S) is the Cayley graph of G with respect to S, and let CG(x) is the centralizer of an element x in G and x is the orbit of x in G. In this paper, we prove that if G act on G by conjugation, the vertex induced subgraph with vertex set CG(x) of the Cayley graph Cay(G, x) is either Hamiltonian or it contain Hamiltonian decompositions. But if n is prime, it is always Hamiltonian. [ABSTRACT FROM AUTHOR]

All connected Cayley graphs over Abelian groups are Hamiltonian. However, for Cayley graphs over non-Abelian groups, Chen and Quimpo prove in [2] that Cayley graphs over Hamiltonian groups (i.e, non- Abelian groups in which every subgroup is normal) are Hamiltonian. In this paper we discuss a few of the ideas which have been developed to establish the existence of Hamiltonian cycles and paths in the vertex induced subgraphs of Cayley graphs over non-Abelian groups. [ABSTRACT FROM AUTHOR]

The hamiltonian cycles and paths in Cayley graphs naturally arise in computer science in the study of word hyperbolic groups and automatic groups. All Cayley graphs over abelian groups are always hamiltonian. However , for Cayley graphs over non-abelian groups, Chen and Quimpo prove in [1] that Cayley graphs over group of order pq, where p and q primes are Hamiltonian and in [2] that Cayley graphs over hamiltonian groups (i.e., non-abelian groups in which every subgroup is normal) are always hamiltonian. In this paper we investigate the existence of complete hamiltonian cycles and hamiltonian paths in the vertex induced subgraphs of Cayley graphs over non-abelian groups. [ABSTRACT FROM AUTHOR]