One and Two-Variable Interlace Polynomials: A Spectral Interpretation

Abstract

Abstract. We relate the one-and two-variable interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions, provide new one and two-variable interlace polynomials, and propose a generalisation of the interlace polynomial to hypergraphs. We also prove conjectures from [15] and equate certain spectral metrics with various evaluations of the interlace polynomial.

Nega–Hadamard Transform, Bent and Negabent Functions

“…In the nega-Hadamard transform context, the basic idea of this result is explained in [5] and equation (15) of [13]. In Lemma 2 we are able to use Hadamard transform because unlike the definition in [5,13] our negacrosscorrelation does not include the factor (−1) wt(u) .…”

“…Substituting z = 0 in the equation (4), we obtain a proof of this fact for the particular case of nega-Hadamard transforms. An equivalent result is proved after equation (15) in [13], and in [11,Theorem 2] …”

“…Towards a nega-periodic analogue of the bent criteria, one can use nega-Hadamard transform and investigate Boolean functions with nega flat spectrum. This motivated several works in the area of Boolean functions [11,13,14,19] in the last few years. In this paper we concentrate on the nega-Hadamard transform in more details.…”

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

“…In this paper, we consider the multivariate interlace polynomial C(G) defined by Courcelle [Cou08] (see Definition 2.1 below) as it subsumes the two-variable interlace polynomial of Arratia, Bollobás, and Sorkin [ABS04b] and the weighted versions of Traldi [Tra08], as well as the interlace polynomials defined by Aigner and van der Holst [AvdH04]. Furthermore, the interlace polynomials Q(x, y) and Q HN n , which have emerged from a spectral view on the interlace polynomials [RP06], are also special cases of Courcelle's multivariate interlace polynomial.…”

Negabent Functions in the Maiorana–McFarland Class

“…More precisely, we construct a subclass of the Maiorana-McFarland class of bent functions in which all functions are also negabent. This construction generalizes the construction of quadratic bent-negabent functions described in [4,5]. These functions are in 2mn variables and have algebraic degree at most n, where m > 1.…”