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

TL;DR

This paper investigates the minimum spectral radius of connected graphs with fixed order and size, confirming Hong's conjecture that such graphs are nearly regular and identifying the minimizers in several cases.

Contribution

It provides a positive answer to Hong's 1993 conjecture for various graph parameters and determines the graphs with minimum spectral radius in multiple cases.

Findings

Hong's conjecture holds for various values of n and e.

Identified graphs with minimum spectral radius in several cases.

Confirmed that minimizer graphs are almost regular.

Abstract

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.