In this paper we compute the exact 2-divisibility of exponential sums associated to elementary symmetric Boolean functions. Our computation gives an affirmative answer to most of the open boundary cases of Cusick-Li-Stǎnicǎ's conjecture. As a byproduct, we prove that the 2-divisibility of these families satisfies a linear recurrence. In particular, we provide a new elementary method to compute 2-divisibility of symmetric Boolean functions.

Abstract. This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between perturbations of two very different symmetric Boolean functions is also included in this work.

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Given a graph G, the tree cover number of the graph, denoted T (G), is the minimum number of vertex disjoint simple trees occurring as induced subgraphs that cover all the vertices of G. This graph parameter was introduced in 2011 as a tool for studying the maximum positive semidefinite nullity of a graph, and little is known about it. It is conjectured that the tree cover number of a graph is at most the maximum positive semidefinite nullity of the graph. In this paper, we establish bounds on the tree cover number of a graph, characterize when an edge is required to be in some tree of a minimum tree cover, and show that the tree cover number of the d-dimensional hypercube is 2 for all d ≥ 2.

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