Upper bounds on the q-spectral radius of book-free and/or K_{s, t}-free graphs

In this paper, we prove two results about the signless Laplacian spectral radius q(G) of a graph G of order n with maximum degree Δ. Let B_n = K₂ +K_n denote a book, i.e., the graph Bn consists of n triangles sharing an edge. The results are the following:. (1) Let 1 ˂ k ≤ l ˂ Δx ˂ n and G be a connected {B_k+1, K_2,l+1}-free graph of order n with maximum degree Δ. Then (Formula Presented) with equality if and only if G is a strongly regular graph with parameters (Δ, k, l). (2) Let s ≥ t≥ 3, and let G be a connected K_s,t-free graph of order n (n≥ s + t). Then (Formula Presented).