Upper bounds on the spectral radius of book-free and/or K2,lK2,l-free graphs

The spectral radius ρ(G)ρ(G) of a graph G   is the largest eigenvalue of its adjacency matrix. Let BkBk denote a book with k   pages. In this paper, we generalize a result of Lu et al [M. Lu, H. Liu, F. Tian, A new upper bound for the spectral radius of graphs with girth at least 5, Linear Algebra Appl. 414 (2006) 512–516.] on the upper bound for the spectral radius of connected graphs with girth at least 5 to connected {Bk+1,K2,l+1}{Bk+1,K2,l+1}-free graphs G   of order nn with maximum degree ΔΔ as follows:ρ(G)⩽[k-l+(k-l)2+4Δ+4l(n-1)]/2with equality if and only if G   is a strongly regular graph with parameters (Δ,k,l)(Δ,k,l). This implies sharp upper bounds for book-free or K2,lK2,l-free graphs.