Some lemmas on spectral radius of graphs: including an application

For a graph $G$, the spectral radius $\rho (G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $\rho (G)$ when $G$ contains a spanning complete bipartite graph. Using these lemmas and typical spectral method, we characterized the unique extremal graph with the maximum spectral radius among all planar graphs of large order $n$ without a cycle of length $\mathrm{\ell }$, where $5\le \mathrm{\ell }\le n$.