On the minimum spectral radius of connected graphs of given order and size

In this article, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices nn and number of edges ee, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this article, we answer Hong’s question positively for various values of nn and ee, and in several cases, we determined the graphs with minimum spectral radius.