Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897656 | Linear Algebra and its Applications | 2018 | 23 Pages |
Abstract
We study the eigenvalues of the unique connected anti-regular graph An. 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=[â1â22,â1+22] contains only the trivial eigenvalues λ=â1 or λ=0, and any closed interval strictly larger than Ω will contain eigenvalues of An 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 An, namely, as n increases the eigenvalues are (approximately) symmetric about the number â12. We also obtain an asymptotic distribution of the eigenvalues as nââ. Finally, the relationship between the eigenvalues of An and the eigenvalues of a general threshold graph is discussed.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Cesar O. Aguilar, Joon-yeob Lee, Eric Piato, Barbara J. Schweitzer,