Spectral measures and dominant vertices in graphs of bounded degree

TL;DR

This paper explores the spectral measures associated with vertices in bounded degree graphs, demonstrating the diverse behaviors of dominance among vertices, including cases with all, some, or no dominant vertices.

Contribution

It characterizes the conditions under which vertices in bounded degree graphs are dominant or not, revealing the full spectrum of possible dominance configurations.

Findings

Vertex-transitive graphs have all vertices dominant.

Some graphs have only a subset of dominant vertices.

Certain graphs lack any dominant vertices.

Abstract

A graph $G = (V, E)$ of bounded degree has an adjacency operator~$A$ which acts on the Hilbert space $\ell^2(V)$. There are different kinds of measures of interest on the spectrum $\Sigma (A)$ of $A$. In particular, each vector $\xi \in \ell^2(V)$ defines a local spectral measure $\mu_\xi$ at $\xi$ on $\Sigma (A)$; therefore each vertex $v \in V$ defines a vector $\delta_v \in \ell^2(V)$ and the associated measure $\mu_v$ on $\Sigma (A)$. A vertex $v$ is dominant if, for all $w \in V$, the measure $\mu_w$ is absolutely continuous with respect to $\mu_v$ (it then follows that, for all $\xi \in \ell^2(V)$, the measure $\mu_\xi$ is absolutely continuous with respect to $\mu_v$). The main object of this paper is to show that all possibilities occur: in some graphs, for example in vertex-transitive graphs, all vertices are dominant; in other graphs, only some vertices are dominant; and there are graphs without dominant vertices at all.