Spectral characterizations of anti-regular graphs

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=[−1−22,−1+22] contains only the trivial eigenvalues λ=−1 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 −12. We also obtain an asymptotic distribution of the eigenvalues as n→∞. Finally, the relationship between the eigenvalues of An and the eigenvalues of a general threshold graph is discussed.