Maximizing the spectral radius of graphs of given size with forbidden a subgraph

Let H₇ denote the $7$-vertex fan graph consisting of a $6$-vertex path plus a vertex adjacent to each vertex of the path. Let K₃ m-3/3K₁ be the graph obtained by joining each vertex of a triangle K₃ to m-3/3 isolated vertices. In this paper, we show that if G is an H₇-free graph with size m≥ 33, then the spectral radius ρ(G)≤ 1+√m-2, equality holds if and only if G K₃ m-3/3K₁ (possibly, with some isolated vertices).