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.

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).

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.

Set X = fM11; M12; M22; M23; M24; Zn; T4n; SD8n; Sz(q); G2(q); V8ng, where M11, M12, M22, M23, M24 are Mathieu groups and Zn, T4n, SD8n, Sz(q), G2(q) and V8n denote the cyclic, dicyclic, semi-dihedral, Suzuki, Ree and a group of order 8n presented by V8n = ha; b j a2n = b4 = e; aba = b􀀀 1; ab􀀀 1a = bi; respectively. In this paper, we compute all eigenvalues of Cay(G; T), where G 2 X and T is minimal, second minimal, maximal or second maximal normal subset of G n feg with respect to its size. In the case that S is a minimal normal subset of G n feg, the summation of the absolute value of eigenvalues, energy of the CAYLEY GRAPH, is evaluated.

Let 􀀀 be a GRAPH with adjacency eigenvalues  1   2  : : :   n. Then the energy of 􀀀 , a concept de ned in 1978 by Gutman, is de ned as E(G) = Σ n i=1 j ij. Also the Estrada index of 􀀀 , which is de ned in 2000 by Ernesto Estrada, is de ned as EE(􀀀 ) = Σ n i=1 e i. In this paper, we compute the eigenvalues, energy and Estrada index of CAYLEY GRAPHs on generalized dihedral groups. As an application, we compute these items for honeycomb toroidal GRAPHs and CAYLEY GRAPHs on dihedral groups.