Spectral radius of graphs forbidden $C_7$ or $C_6^{\mathrm{△}}$

Let $C_k^{\mathrm{△}}$ be the graph obtained from a cycle $C_k$ by adding a new vertex connecting two adjacent vertices in $C_k$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\mathrm{△}}$. As a byproduct, we will show that if the spectral radius $\lambda (G)\ge 1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\mathrm{\nabla }\left(\frac{m-3}{3}{K}_1\right)$.