The spectral Turán problem about graphs of given size with forbidden subgraphs

Let G be a graph of size m and [Formula: see text] be the spectral radius of its adjacency matrix. A graph is said to be H-free if it does not contain a subgraph isomorphic to H. Let [Formula: see text] be the graph obtained by adding a pendent vertex to a cycle of length l. In this paper, we determine the graphs with the maximum spectral radius over [Formula: see text]-free graphs, and [Formula: see text]-free graphs with [Formula: see text].