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 dominating set DÍV of a graph G= (V; E) is said to be a Connected cototal dominating set if (D) is Connected and (V-D) ¹f, contains no isolated vertices. A Connected cototal dominating set is said to be minimal if no proper subset of D is Connected cototal dominating set. The Connected cototal domination number (G) of G is the minimum cardinality of a minimal Connected cototal dominating set of G. In this paper, we begin an investigation of Connected cototal domination number and obtain some interesting results.

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.

Introduction: Let G be a simple undirected graph over the vertex set V. Let I(G) denotes the edge ideal of G and DG be the simplicial complex whose faces correspond to the independent sets of G. This simplicial complex reflects many nice properties of G. A simplicial complex D is called shellable if the facets can given a linear order F1,…,Ft such that for all 1£t<f£s, there exists some vÎF1\Ft and some LÎ{1,…,f-1} with F1\Ft={v}. A result due to Hochster says that every pure shellable complex is Cohen-Macaulay over every field. A graph is called shellable, if the simplicial complex DG is a shellable simplicial complex.Aim: In this paper we focus on the question of what graphs G have the property that `G is Cohen-Macaulay, i.e. R/I(`G) is Cohen-Macaulay. We prove that the complement of a Connected triangle-free graph is pure shellable and consequently Cohen-Macaulay.Methods: By providing an explicit shelling for the facets of D`G, whenever G is a Connected triangle-free graph, we show `G, the complement of G, is pure shellable and consequently Cohen-Macaulay.Conclusion: The complement of any Connected bipartite graph and any cycle is pure shellable and hence Cohen-Macaulay.

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.