Spectral characterizations of anti-regular graphs

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 intervalcontains 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.2000 Mathematics Subject Classification. Primary 05C50, 15B05; Secondary 05C75, 15A18.

Eigenvalue-free interval for threshold graphs

“…It follows that threshold graphs have no non-trivial eigenvalues in the interval (−1 − √ 2)/2, (−1 + √ 2)/2 . These results confirm two conjectures of [1] and improve the aforementioned result of [8].…”

“…In [1] it is shown that for any n, η − (A n ) < (−1 − √ 2)/2 and (−1 + √ 2)/2 < η + (A n ). This result and Theorem 6 imply the following corollary follows.…”

The role of the anti-regular graph in the spectral analysis of threshold graphs

“…Using a similar eigenvalue interlacing technique, Corollary 4.2 was proved by E. Ghorbani [9]. The following conjecture was made in [1]. Conjecture 4.1.…”

“…The graph A n is a threshold graph with binary string b = 0101 · · · 01 when n is even and b = 00101 · · · 01 when n is odd. It was proved in [18] (see also [1]) that A n has simple eigenvalues and moreover has inertia i(A 2k ) = (k, 0, k) if n = 2k is even and i(A 2k+1 ) = (k, 1, k) if n = 2k + 1 is odd. Thus, λ k+1 (A 2k+1 ) = 0 and A 2k+1 does not contain −1 as an eigenvalue.…”

“…In [2], the authors investigate the normalized adjacency eigenvalues and energy of threshold graphs and they obtain results that parallel the known results for the adjacency eigenvalues. In [1], a nearly complete characterization of the eigenvalues of anti-regular graphs is given. Specifically, it is proved that no anti-regular graph has an eigenvalue in the interval Ω = [ −1− derive an explicit expression for the characteristic polynomial of a threshold graph and use it to find the determinant of A(G) and prove that no two non-isomorphic graphs are cospectral.…”