The paper begins with a simple circular lock problem that shows how the Combinatorial Nullstellensatz relates to the discrete Fourier Transform.Specifically, the lock shows a relationship between detecting perfect matchings in bipartite graphs using the Combinatorial Nullstellensatz and detecting a maximum rank independent set in the intersection of two matroids in the Fourier transform of a specially chosen function. Finally, an application of the uncertainity principle computes a lower bound for the product of perfect matchings and the number of independent sets.

A \({magic\; square}\) of order \(n\) is an \(n \times n\) array of integers from \(1, 2, \ldots, n^2\) such that the sum of the integers in each row, column, and diagonal is the same number. Two magic squares are \({equivalent}\) if one can be obtained from the other by rotation or reflection. The \({complement}\) of a magic square \(M\) of order \(n\) is obtained by replacing every entry \(a\) with \(n^2 + 1 – a\), yielding another magic square. A magic square is \({self-complementary}\) if it is equivalent to its complement. In this paper, we prove a structural theorem characterizing self-complementary magic squares and present a method for constructing self-complementary magic squares of even order. Combining this construction with the structural theorem and known results on magic squares, we establish the existence of self-complementary magic squares of order \(n\) for every \(n \geq 3\).

Let \(G\) be a graph on \(n\) vertices. If for any ordered set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\), where the vertices in \(S\) appear in the sequence order \(v_1, v_2, \ldots, v_k\), there exists a \(v_1-v_k\) (Hamiltonian) path containing \(S\) in the given order, then \(G\) is \(k\)-ordered (Hamiltonian) connected. In this paper, we show that if \(G\) is \((k+1)\)-connected and \(k\)-ordered connected, then for any ordered set \(S\), there exists a \(v_1-v_k\) path \(P\) containing \(S\) in the given order such that \(|P| \geq \min\{n, \sigma_2(G) – 1\}\), where \(\sigma_2(G) = \min\{d_G(u) + d_G(v) : u,v \in V(G); uv \notin E(G)\}\) when \(G\) is not complete, and \(\sigma_2(G) = \infty\) otherwise. Our result generalizes several related results known before.

Let \(G\) be a simple graph. The incidence energy ( \(IE\) for short ) of \(G\) is defined as the sum of the singular values of the incidence matrix. In this paper, a new lower bound for \(IE\) of graphs in terms of the maximum degree is given. Meanwhile, an upper bound and a lower bound for \(IE\) of the subdivision graph and the total graph of a regular graph \(G\) are obtained, respectively.