On strong nodal domains for eigenfunctions of Hamming graphs

This paper aims to compute and estimate the eigenvalues of the Hodge Laplacians on directed graphs. We have devised a new method for computing Hodge spectra with the following two ingredients. (I) We have observed that the product rule does work for the so-called normalized Hodge operator, denoted by $\Delta _{p}^{(a)},$ where $a$ refers to the weight that is used to redefine the inner product in the spaces $\Omega _{p}$. This together with the K\"{u}nneth formula for produ...

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$ and $f(v)\equiv d_H(v) \pmod{2}$. We prove sharp upper bounds for certain eigenvalues in an $h$-edge-connected graph $G$ with given minimum degree to guarantee the existence of a $(g,...

This paper is concerned with the lower bounds for the principal frequency of the $p$-Laplacian on $n$-dimensional Euclidean domains. In particular, we extend the classical results involving the inner radius of a domain and the first eigenvalue of the Laplace operator to the case $p\neq2$. As a by-product, we obtain a lower bound on the size of the nodal set of an eigenfunction of the $p$-Laplacian on planar domains.

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real cons...

In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2\pi\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2\pi\mathbb{Z})\times(\mathbb{R}/2\rho\pi\mathbb{Z})\,$, with $\rho=\frac{1}{\sqrt{3}}\,$, and const...

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $\nu_n$ of the $n$-th eigenfunction satisfies $n\ge \nu_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-\nu_n$ in the case...

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's re...

The well-known middle levels problem is to find a Hammiltonian cycle in the graph induced from the binary Hamming graph $\cH_2(2k+1)$ by the words of weight $k$ or $k+1$. In this paper we define the $q$-analog of the middle levels problem. Let $n=2k+1$ and let $q$ be a power of a prime number. Consider the set of $(k+1)$-dimensional subspaces and the set of $k$-dimensional subspaces of $\F_q^n$. Can these subspaces be ordered in a way that for any two adjacent subspaces $X$ a...

This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our estimates hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts.

In this paper, we determine the set of all distinct eigenvalues of the line graph which is induced by the first and second layers of the hypercube $ Q_n $, $n>3$. We show that this graph has precisely five distinct eigenvalues and all of its eigenvalues are integers