On the limit points of the smallest eigenvalues of regular graphs

TL;DR

This paper explores the range of smallest eigenvalues in regular graphs, providing infinite examples in specific intervals, identifying limit points, and characterizing extremal graphs with eigenvalues less than -2.

Contribution

It introduces infinite families of regular graphs with smallest eigenvalues in particular intervals and determines key limit points and extremal graphs for eigenvalues below -2.

Findings

Identifies infinite examples of regular graphs with eigenvalues in specific intervals.

Determines the largest and second largest limit points of smallest eigenvalues less than -2.

Characterizes the unique 3-regular graph with the maximum smallest eigenvalue below -2.

Abstract

In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[\alpha_1, -1-\sqrt2)$ where $\alpha_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than -2. Moreover we determine the supremum of the smallest eigenvalue among all connected 3-regular graphs with smallest eigenvalue less than -2 and we give the unique graph with this supremum value as its smallest eigenvalue.