Let n be any positive integer, the friendship GRAPH Fn consists of n edge-disjoint triangles that all of them meeting in one vertex. A GRAPH G is called cospectral with a GRAPH H if their adjacency matrices have the same eigenvalues. Recently in http://arxiv.org/pdf/1310.6529v1.pdf it is proved that if G is any GRAPH cospectral with Fn (n ¹16), then G=Fn. Here we give a proof of a special case of the latter: Any CONNECTED GRAPH cospectral with Fn is isomorphic to Fn. Our proof is independent of ones given in http://arxiv.org/pdf/1310.6529v1.pdf and the proofs are based on our recent results given in [Trans. Comb., 2 no. 4 (2013) 37-52.] using an upper bound for the largest eigenvalue of a CONNECTED GRAPH given in [J. Combinatorial Theory Ser. B 81 (2001) 177-183.].

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.

If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a CONNECTED GRAPH G, the generalized adjacency matrix A (G) is de ned as A (G) =  D(G) + (1  ) A(G), where 0    1...

The divisibility GRAPH D (G) for a finite group G is a GRAPH with vertex set cs (G) \ {1} where cs (G) is the set of conjugacy class sizes of G. Two vertices a and b are adjacent whenever a divides b or b divides a. In this paper we will find the number of CONNECTED components of D (G) where G is a simple Zassenhaus group or an sporadic simple group.

The order GRAPH of a group G, denoted by 􀀀  (G), is a GRAPH whose vertices are non trivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if jHj   jKj or jKj   jHj. In this paper, we study the connectivity and diameter of this GRAPH. Also we give a relation between the order GRAPH and prime GRAPH of a group.

In this article, we study connections between components of the Cayley GRAPH Cay(G, A), where A is an arbitrary subset of a group G, and cosets of the subgroup of G generated by A. In particular, we show how to construct generating sets of G if Cay(G, A) has finitely many components. Furthermore, we provide an algorithm for finitely minimal generating sets of finite groups using their Cayley GRAPHs.