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

Abstract

The purpose of this paper is to highlight the role played by the anti-regular graph within the class of threshold graphs. Using the fact that every threshold graph contains a maximal antiregular graph, we show that some known results, and new ones, on the spectral properties of threshold graphs can be deduced from (i) the known results on the eigenvalues of anti-regular graphs, (ii) the subgraph structure of threshold graphs, and (iii) eigenvalue interlacing. In particular, we prove that no threshold graph contains an eigenvalue in the interval Ω =], except possibly the trivial eigenvalues −1 and/or 0, determine the inertia of a threshold graph, and give partial results on a conjecture regarding the optimality of the non-trivial eigenvalues of an anti-regular graph within the class of threshold graphs.

A survey of antiregular graphs

“…Theorem 2.3 states that that every tree of order n is isomorphic to a subgraph of the connected antiregular graph A n . Aguilar et al [2] showed that a similar property of the connected antiregular graphs holds also for the class of threshold graphs.…”

“…In [2], the authors reported some partial results towards Conjecture 5.1 -more precisely, they prove that this conjecture is true for all threshold graphs of order n except for n − 2 critical cases, and these cases are considered in a recent preprint [60]. Thus, combining the results obtained in [2,60], we can say that Conjecture 5.1 holds. Theorem 5.12.…”

“…Proposition 5.1. [2] If n ≥ 3 is odd, then µ − (A n ) ≥ µ − (A n+1 ) and µ + (A n ) ≥ µ + (A n+1 ). Also, if n ≥ 4 is even, then µ − (A n ) ≤ µ − (A n+1 ) and µ + (A n ) ≤ µ + (A n+1 ).…”

“…Theorem 2.6. [2] Every connected threshold graph of order n ≥ 2 is isomorphic to an induced subgraph of the connected antiregular graph A 2n−2 . In fact, let G be a connected threshold graph of order n and with binary string b = 0…”

On the Seidel spectrum of threshold graphs

“…Since last decade investigation on the spectral properties of adjacency eigenvalues gained lot of attention. We found lot of papers in this direction [1,2,4,6,10,11,12,13,17]. Bapat [4] proved that the number of negative, zero, and positive eigenvalues of a threshold graph can be find out directly from its binary representation.…”