The purpose of this paper is the study of Cayley Graph associated to a semihy-pergroup(or hypergroup). In this regards , rst we associate a Cayley Graph to every semi-hypergroup and then we study the properties of this Graph, such as Hamiltonian cycles in this Graph. Also, by some of examples we will illustrate the properties and behavior of these Cayley Graphs, in particulars we show that the properties of a Cayley Graph associated to a semihypergroup is completely di, erent with respect to the Cayley Graph associated to a semigroup(group). Also, we brie y discuss on category of Cayley Graphs associated to semi-hypergroups and construct a functor from this category to the category of diGraphs. Finally, we give an application the Cayley Graph of a hypergroupoid to a social network.

In this paper we introduce mixed unitary Cayley Graph Mn (n > 1) and compute its eigenvalues. We also compute the energy of Mn for some n.

Let G(V, E) be a Graph. The common neighborhood Graph (conGraph) of G is a Graph with vertex set V, in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of conGraphs under Graph operations; Graph union, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley Graphs and its conGraphs.

In this paper, we introduce a suitable generalization of Cayley Graphs that is de ned over polygroups (GCP-Graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known Graph operations and apply to GCP-Graphs. Finally, we prove that the product of GCP-Graphs is again a GCP-Graph.

In this paper, first we introduce the end of locally finite Graphs as an equivalence class of infinite paths in the Graph. Then we mention the ends of finitely generated groups using the Cayley Graph. It was proved that the number of ends of groups are not depended on the Cayley Graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For each of these numbers, some results have been obtained in the structure of groups, the most well-known of which is Stallings theorem providing the structure of groups with more one ends as the amalgamated free product or HNN extension. Specifically, it was proved that group with exactly two ends is a virtually Z group. After that, we introduce the trivial end of the Graphs and show that the trivial end is exactly the same as the special type of infinite path. Finally, we prove that the existence of trivial end for Cayley Graph of a group is equivalent to being a free group, and this implies that the Cayley Graph of a group has a trivial end if and only if all of its ends are trivial.

A Graph is called integral if all eigenvalues of its adjacency matrix are integers. Given a subset S of a  nite group G, the bi-Cayley Graph BCay(G; S) is a Graph with vertex set G  f1; 2g and edge set ff(x; 1); (sx; 2)g j s 2 S; x 2 Gg. In this paper, we classify all  nite groups admitting a connected cubic integral bi-Cayley Graph.

LetR be a finite commutative ring. Let Z (R) and J (R) be the set of all zero-divisor elements and the Jacobson radical of R, respec-tively. The zero-divisor Cayley Graph of R, denoted by Z CAY (R), is the Graph obtained by setting all the elements of Z (R) to be the vertices and defining distinct verticesx and y to be adjacent if and only if x Îy 2 Z (R).The induced subGraph of Z CAY (R) on the vertex set Z (R)\ J (R) is de-noted byZ CAY* (R). In this paper, the basic properties of Z CAY (R) and ZCAY* (R) are investigated and some characterization results regarding connectedness, girth and planarity of Z CAY (R) and Z CAY* (R) are given. Finally, we study the clique number of Z CAY (R).

The unitary addition Cayley Graph Gn[ω, ] of Eisenstein integers modulo n has the vertex set En[ω, ], the set of Eisenstein integers modulo n. Any two vertices x = a1 + ω, b1, y = a2 + ω, b2 of Gn[ω, ] are adjacent if and only if gcd(N(x + y), n) = 1, where N is the norm of any element of En[ω, ] given by N(a + ω, b) = a2 + b2 −,ab. In this paper we obtain some basic Graph invariants such as degree of the vertices, number of edges, diameter, girth, clique number and chromatic number of unitary addition Cayley Graph of Eisenstein integers modulo n. This paper also focuses on determining the independence number of the above mentioned Graph.

We call a Cayley Graph G= Cay (G, S) normal for G, if the right regular representation R (G) of G is normal in the full automorphism group of Aut (G). In this paper, a classification of all non-normal Cayley Graphs of finite abelian group with valency 6 was presented.