Normalized Laplacian spectrum of two new types of join graphs

Let G be a graph without an isolated vertex, the normalized Laplacian matrix ~ L(G) is dened as ~ L(G) = D1 2 L(G)D12 , where D is a diagonal matrix whose entries are degree of vertices of G. The eigenvalues of ~ L(G) are called as the normalized Laplacian eigenvalues of G. In this paper, we obtain the normalized Laplacian spectrum of two new types of join graphs. In continuing, we determine the integrality of normalized Laplacian eigenvalues of graphs. Finally, the normalized Laplacian energy and degree Kirchhoindex of these new graph products are derived