Maximizing the spectral radius of graphs of given size with a forbidden 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 [Formula presented] be the graph obtained by joining each vertex of a triangle K₃ to [Formula presented] 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 [Formula presented] (possibly, with some isolated vertices).