Search Results

Let $\Gamma$ be an undirected graph the degree of whose $n$ vertices is $\leq D$ without necessarily being constant. Say we have bounds of type $\leq \gamma^{2 k}$ for the number of closed geodesics o …

Let $A$ be the adjacency operator of a symmetric graph $\Gamma$. (It may be weighted and/or non-regular, but, to keep it simple, let us say it is unweighted and regular of degree $d$.) We want to boun …

Let $\Gamma$ be a graph. Write $A$ for its adjacency matrix. It is trivial to show that $\mathrm{Tr} A^k$ equals the number of closed walks of length k, that is, the number of $k$-step walks that retu …

Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the numb …

Very basic and somewhat open-ended question: Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i …

Let $\Gamma$ be an undirected finite graph. Write $M_r$ for the number of non-backtracking closed walks of length $r$ in $\Gamma$, i.e., walks where a step is never followed by its inverse. Let $A$ be …

Let $\Gamma=(V,E)$ be an undirected graph of degree $d$. (Say $d$ is a large constant and the number of vertices $n=|V|$ is much larger.) Let $W_0$ be the space of functions $f:V\to \mathbb{C}$ with a …

Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by $$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$ (Alternati …

Let $\Gamma$ be an undirected graph of bounded degree $d$ with $V = \{1,2,\dotsc,N\}$ as its set of vertices, and edges only between vertices that are at a distance $\leq M$ apart (where $M$ is much s …

Consider a $d$-regular, $k$-uniform hypergraph: the elements $S$ of its set $E$ of edges are subsets of $V$ of size $k$, and each vertex $v\in V$ is in $d$ edges. We can then define its adjacency tens …

Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that $$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$ for any $k\geq 1$. …