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.

翻译：本文研究了Hong于1993年提出的一个关于给定阶数与边数的连通图的邻接矩阵最小谱半径的问题。Hong询问：在所有具有给定顶点数$n$与边数$e$的连通图中，是否谱半径最小（即最小化图）的图必须是几乎正则的，这意味着其最大度与最小度之差至多为1。本文针对多种$n$和$e$的取值对Hong的问题给出了肯定的回答，并在若干情形下确定了具有最小谱半径的图。