The maximum spectral radius of graphs of given size with forbidden subgraph.

Let G be a graph of size m and ρ (G) be the spectral radius of its adjacency matrix. A graph is said to be F -free if it does not contain a subgraph isomorphic to F. Let θ 1 , 2 , 3 denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths 1, 2, 3, respectively. Recently, Li, Sun and Wei showed that for any θ 1 , 2 , 3 -free graph G of size m ≥ 8 , ρ (G) ≤ 1 + 4 m − 3 2 , with equality if and only if G ≅ S m + 3 2 , 2 , where S m + 3 2 , 2 = K 2 ▿ K m − 1 2 ‾. However, this bound is not attainable when m is even. In this paper, we characterize the graphs with the maximum spectral radius over K 2 , r + 1 -free non-star graphs with m ≥ (4 r + 2) 2 + 1 , the graphs with the maximum spectral radius over θ 1 , 2 , 3 -free graphs with m ≥ 22 when m is even, and the graphs with the second largest spectral radius over θ 1 , 2 , 3 -free graphs with m ≥ 22 when m is odd. [ABSTRACT FROM AUTHOR]