Classification of graphs by Laplacian eigenvalue distribution and independence number

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