Spectral Properties of Boolean Functions, Graphs and Graph States

Generalisations of the bent property of a Boolean function are presented, by proposing spectral analysis of the Boolean function with respect to a well-chosen set of local unitary transforms. Quadratic Boolean functions are related to simple graphs and it is shown that the orbit generated by some graph transforms can be found within the spectra of certain unitary transform sets. The flat spectra of a quadratic Boolean function with respect to those transforms are related to modified versions of its associated adjacency matrix. The flat spectra of concrete recursive structures are found using this method. We derive a spectral interpretation of the interlace polynomials (in one or two variables) of a graph and we relate to one of them a quantum measure of entanglement of the associated quantum state. We characterise the values of the spectra of a quadratic Boolean function. We give a formula for the weight hierarchy for a binary linear code in terms of a modified interlace polynomial. We derive as well a spectral interpretation of the pivot operation on a graph and generalise this operation to hypergraphs. We enumerate the number of inequivalent pivot orbits for small numbers of vertices. We also construct a family of Boolean functions of degree higher than two with a large number of flat spectra with respect to the {I,H}n set of transforms, and compute a lower bound on this number. We show how to change the degree of a Boolean function via pivot operation. Finally, we give concrete formulae for the spectra of a wide range of vectors with respect to well-chosen transforms sets.