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

Abstract The spectral radius ρ ( G ) of a graph G is the largest eigenvalue of its adjacency matrix. Let B k 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 { B k + 1 , K 2 , l + 1 } -free graphs G of order n with maximum degree Δ as follows: ρ ( G ) ⩽ [ k - l + ( k - l ) 2 + 4 Δ + 4 l ( n - 1 ) ] / 2 with equality if and only if G is a strongly regular graph with parameters ( Δ , k , l ) . This implies sharp upper bounds for book-free or K 2 , l -free graphs.