On the maximum spectral radius of connected graphs with a prescribed order and size

The spectral radius of a graph is the largest modulus of an eigenvalue of its adjacency matrix. Let ${\mathcal{C}}_{n,e}$ be the set of all the connected simple graphs with $n$ vertices and $n-1+e$ edges. Here, we solve the spectral radius maximization problem on ${\mathcal{C}}_{n,e}$ when $e\le 130$ or $n\ge e+2+13\sqrt{e}$.