Spectral characterizations of anti-regular graphs

TL;DR

This paper characterizes the eigenvalues of the connected anti-regular graph using Chebyshev polynomials, revealing their distribution, bounds, symmetry, and relationship to threshold graphs.

Contribution

It provides a trigonometric equation for eigenvalues, interval bounds, and asymptotic distribution analysis, advancing understanding of anti-regular graph spectra.

Findings

Interval $ frac{-1-\sqrt{2}}{2}$ to $ frac{-1+\sqrt{2}}{2}$ contains only trivial eigenvalues

Eigenvalues are symmetric about -1/2 as n increases

Asymptotic eigenvalue distribution as n approaches infinity

Abstract

We study the eigenvalues of the unique connected anti-regular graph $A_n$. Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an almost complete characterization of the eigenvalues. In particular, we show that the interval $\Omega=[\tfrac{-1-\sqrt{2}}{2}, \tfrac{-1+\sqrt{2}}{2}]$ contains only the trivial eigenvalues $\lambda = -1$ or $\lambda=0$, and any closed interval strictly larger than $\Omega$ will contain eigenvalues of $A_n$ for all $n$ sufficiently large. We also obtain bounds for the maximum and minimum eigenvalues, and for all other eigenvalues we obtain interval bounds that improve as $n$ increases. Moreover, our approach reveals a more complete picture of the bipartite character of the eigenvalues of $A_n$, namely, as $n$ increases the eigenvalues are (approximately) symmetric about the number $-\tfrac{1}{2}$. We also obtain an asymptotic distribution of the eigenvalues as $n\rightarrow\infty$. Finally, the relationship between the eigenvalues of $A_n$ and the eigenvalues of a general threshold graph is discussed.