  In this paper, 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 $n$ and number of edges $e$, 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 paper, we answer Hong's question positively for various values of
$n$ and $e$ and in several cases, we determined the graphs with minimum
spectral radius.
