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

In this paper, two results about the signless Laplacian spectral radius q(G) of a graph G of order n with maximum degree ∆ are proved. Let Bn = K2 + Kn 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 < ∆ < n and G be a connected {Bk+1,K2,l+1}-free graph of order n with maximum degree ∆. Then q(G) ≤ 1 4 [ 3∆ + k − 2l + 1 + √ (3∆ + k − 2l + 1)2 + 16l(∆ + n− 1) ] with equality if and only if G is a strongly regular graph with parameters (∆, k, l). (2) Let s ≥ t ≥ 3, and let G be a connected Ks,t-free graph of order n (n ≥ s + t). Then q(G) ≤ n + (s− t + 1)1/tn1−1/t + (t− 1)(n− 1)1−3/t + t− 3.