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.

The Birkhoff graph B n is the Cayley of symmetric group S , where two permutations are adjacent if they differ by a single cycle. Our main result tighter upper bound on independence number ? ( ) namely, we show that ? O ! / 1 . 9 7 improving previous known 2 Kane et al. (2017). approach combines higher-order version their representation theoretic techniques with linear programming. With an expl...