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

Abstract

Let $H_7$ denote the 7-vertex fan graph consisting of a 6-vertex path plus a vertex adjacent to each vertex of the path. Let $K_3\vee \frac{m-3}{3}{K}_1$ be the graph obtained by joining each vertex of a triangle $K_3$ to $\frac{m-3}{3}$ isolated vertices. In this paper, we show that if G is an $H_7$-free graph with size $m\ge 33$, then the spectral radius $\rho \left(G\right)\le 1+\sqrt{m-2}$, equality holds if and only if $G\cong K_3\vee \frac{m-3}{3}{K}_1$ (possibly, with some isolated vertices).