Classification of graphs by Laplacian eigenvalue distribution and independence number

Let mGI denote the number of Laplacian eigenvalues of a graph G in an interval I and let α(G) denote the independence number of G. In this paper, we determine the classes of graphs that satisfy the condition mG[0, n − α(G)] = α(G) when α(G) = 2 and α(G) = n−2, where n is the order of G. When α(G) = 2, G ∼= K1∇Kn−m∇Km−1 for some m ≥ 2. When α(G) = n − 2, there are two types of graphs B(p, q, r) and B′(p, q, r) of order n = p + q + r + 2, which we call the binary star graphs. Also, we show that the binary star graphs with p = r are determined by their Laplacian spectra.