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 $n \ge e + 2 + 13\sqrt{e}$.