From the eigenvalue equation we obtained by using Chebyshev polynomial of the second kind, we deduce three functions in terms of An. Through analyzing the functions, we discovered that the interval Ω=[(-1-√2)/2, (-1+√2)/2] contains only the trivial eigenvalues 0 and -1. We also figured out that as n increases, the eigenvalues of An become almost symmetric about the number -1/2. Finally, we conjectured that the eigenvalue- free interval bound Ω does not contain an eigenvalues of any threshold graph other than 0 and -1, and that among all threshold graphs on n vertices, the antiregular graph An has the eigenvalues closest to the boundary points of Ω.
Piato, Eric and Lee, Joon-yeob
"Spectral Characterization of Anti-Regular Graphs,"
Proceedings of GREAT Day: Vol. 2019
, Article 10.
Available at: https://knightscholar.geneseo.edu/proceedings-of-great-day/vol2019/iss1/10