Related papers: Towards a Calculus for Non-Linear Spectral Gaps [E…

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…

A graph $G(V,E)$ is a threshold graph if there exist non-negative reals $w_v, v \in V$ and $t$ such that for every $U \subseteq V$, $\sum_{v \in U} w_v\leq t$ if and only if $U$ is a stable set. The {\it threshold dimension} of a graph…

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y…

We analyze the spectral properties of the high-dimensional random geometric graph $G(n, d, p)$, formed by sampling $n$ i.i.d vectors $\{v_i\}_{i=1}^{n}$ uniformly on a $d$-dimensional unit sphere and connecting each pair $\{i,j\}$ whenever…

In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the…

We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define $G(p)$ to be a random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p \in…

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is…

In this paper we study spectra of Laplacians of infinite weighted graphs. Instead of the assumption of local finiteness we impose the condition of summability of the weight function. Such graphs correspond to reversible Markov chains with…

Almost $4$-connectivity is a weakening of $4$-connectivity which allows for vertices of degree three. In this paper we prove the following theorem. Let $G$ be an almost $4$-connected triangle-free planar graph, and let $H$ be an almost…

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\mathbb{R})$ is a symmetric stochastic matrix. Denote the second-largest eigenvalue of…

Let $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $x_1, \dots, x_k$, take $x_{k+1}$ to be any vertex maximizing the sum of distances to the existing vertices and iterate: we…

Graph complements G(n) of cyclic graphs are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian graphs with a Z(n) symmetry, Shannon capacity 2 and known Wiener and Harary index. There is an explicit spectral zeta…

The following problem has been proposed in [Research problems from the Aveiro workshop on graph spectra, {\em Linear Algebra and its Applications}, {\bf 423} (2007) 172-181.]:\\ (Problem AWGS.4) Let $G_n$ and $G'_n$ be two nonisomorphic…

If $R$ is a commutative ring, $M$ a compact $R$-oriented manifold and $G$ a finite graph without loops or multiple edges, we consider the graph configuration space $M^G$ and a Bendersky-Gitler type spectral sequence converging to the…

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space $X$ coarsely (resp. uniformly) embeds into a Banach space $Y$…

Given a finite simple graph G, let G' be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If L(0)=0<L(1) <=…

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set…

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $\lambda(G) \leq h(G)$, where…

The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalue. The study of graph-related matrices has attracted considerable attention, leading to a substantial body of findings. In this…

We study the least doubling constant among all possible doubling measures defined on a (finite or infinite) graph $G$. We show that this constant can be estimated from below by $1+ r(A_G)$, where $r(A_G)$ is the spectral radius of the…