1. Spectral characterizations of anti-regular graphs.
- Author
-
Aguilar, Cesar O., Lee, Joon-yeob, Piato, Eric, and Schweitzer, Barbara J.
- Subjects
- *
REGULAR graphs , *CHEBYSHEV polynomials , *EIGENVALUES , *GRAPH theory , *MATRICES (Mathematics) , *ASYMPTOTIC distribution - 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 Ω = [ − 1 − 2 2 , − 1 + 2 2 ] contains only the trivial eigenvalues λ = − 1 or λ = 0 , and any closed interval strictly larger than Ω 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 − 1 2 . We also obtain an asymptotic distribution of the eigenvalues as n → ∞ . Finally, the relationship between the eigenvalues of A n and the eigenvalues of a general threshold graph is discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF