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Let G be a finite group and cd (G) be the set of nonlinear irreducible CHARACTER degrees of G. Suppose that  (G) denotes the set of primes dividing some element of cd (G). The bipartite divisor GRAPH for the set of CHARACTER degrees which is denoted by B(G), is a bipartite GRAPH whose vertices are the disjoint union of  (G) and cd (G), and a vertex p 2  (G) is connected to a vertex a 2 cd (G) if and only if p|a. In this paper, we investigate the structure of a group G whose GRAPH B(G) has five vertices. Especially we show that all these groups are solvable.

Let G be a finite group, and Irr(G) be the set of complex irreducible CHARACTERs of G. Let  (G) be the set of prime divisors of CHARACTER degrees of G. The CHARACTER degree GRAPH of G, which is denoted by Δ (G), is a simple GRAPH with vertex set  (G), and we join two vertices r and s by an edge if there exists a CHARACTER degree of G divisible by rs. In this paper, we prove that if G is a finite group such that Δ (G) = Δ (PSL2(q)) and jGj = jPSL2(q)j, then G  = PSL2(q).

Let G be a , nite group, Irr(G) be the set of complex irreducible CHARACTERs of G, and denote by cd(G) = f, (1): ,2 Irr(G)g the set of irreducible CHARACTER degrees of G. Classifying , nite groups by the properties of their CHARACTERs is an interesting problem in representation theory. In [3], Huppert conjectured that each , nite non-abelian simple group G is CHARACTERized by the set cd(G): In [3, 4, 13, 15], it was shown that the conjecture holds for simple groups such as L2(q) and Sz(q). In this paper, we attempt to CHARACTERize the Janko groups J1,J3 and J4 by their orders and one irreducible CHARACTER degree. Also authors guess that the result is not correct for Janko group J2, but they have not found any counterexample yet. Let G be a , nite group,L(G) denotes the largest irreducible CHARACTER degree of G. The following result is our main theorem in the third section.

The independence GRAPH Ind(G) of a GRAPH G is the GRAPH with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence GRAPH of Cartesian product of d copies of complete GRAPHs Kq, which is known as the Hamming GRAPH H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming GRAPH H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.