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.

We investigate the classical H. Zassenhaus conjecture for integral group rings of alternating groups A9 and A10 of degree 9 and 10, respectively. As a consequence of our previous results we con rm the Prime Graph Conjecture for integral group rings of An for all n£10.

In this paper, we define duplication corona, duplication neigh-borhood corona and duplication edge corona of two graphs. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As an application, our results enable us to construct infinitely many pairs of cospectral graphs and also integral graphs.

In this paper we specify the class of integral circulant graphs ICG(n; D), which can be characterized by their order n and the set D of positive divisors of n in such a way that they have the vertex set Zn and the edge set (Formola). This group of graphs is called BMM graphs because of the form of its set of edges. A bipartite G graph is a graph whose vertex set can be divided into two subsets X, and Y such that no two vertices in X and no two vertices in Y are adjacent. The duplicate graph is called complete if each vertex in X is connected to all vertices in Y. This graph is represented by K_ (m, n), if | X | = m and | Y | = n. Multipartite graphs are also defined as bipartite graphs.

In this paper, we construct a new class of integral bipartite graphs (not necessarily trees) with large even diameters. In fact, for every nite set A of positive integers of size k we construct an integral bipartite graph G of diameter 2 k such that the set of positive eigenvalues of G is exactly A. This class of integral bipartite graphs has never found before.

A graph is called circulant if it is a Cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. Let D be a set of positive, proper divisors of the integer n > 1. The integral circulant graph ICGn (D) has the vertex set Z n and the edge set E (ICGn (D)) = {{a, b}; gcd (a – b, n) Î D}. Let n = p1p2….pkm, where p1, p2, …., pk are distinct prime numbers and gcd (p1p2…pk, m) = 1. The open problem posed in paper [A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881-1889] about calculating the energy of an arbitrary integral circulant ICGn (D) is completely solved in this paper, where D = {p1, p2, …., pk}.

Let n  1 be an integer. The hypercube Qn is the graph whose vertex set is f0; 1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. In the graph Qn, the layer Lk is the set of vertices with exactly k 1’ s, namely, vertices of weight k, 1  k  n. The hyper-star graph B(n; k) is the subgraph of Qn induced by layers Lk and Lk+1; 0 < k < n. In this paper, we determine the spectrum of the hyperstar graph B(n; k) and L(B(n; k)), where L(B(n; k)) is the line graph of the graph B(n; k). In particular, we show that the graph L(B(n; k)) is an integral graph, that is, all of its eigenvalues are integers. In this paper, we investigate some of the algebraic properties of the graph B(n; k) and its line graph L(B(n; k)). In particular, we determine the spectrum of these graphs.

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.

In this paper, we find the star chromatic number of central graph of complete bipartite graph and corona graph of complete graph with path and cycle.