# Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 215-256.

Spectral measures and dominant vertices in graphs of bounded degree

**Authors**:
Claire Bruchez (1), Pierre de la Harpe (2), Tatiana Nagnibeda (3)

**Author institution:**(1) Section de mathematiques, Universite de Geneve, Uni Dufour,
24 rue du General Dufour, Case postale 64, 1211 Geneve 4, Suisse

(2) Section de mathematiques, Universite de Geneve, Uni Dufour,
24 rue du General Dufour, Case postale 64, 1211 Geneve 4, Suisse

(3) Section de mathematiques, Univ. de Geneve, Uni Dufour,
24 rue du General Dufour, Case postale 64, 1211 Geneve 4, Suisse

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

**DOI: **http://dx.doi.org/10.7900/jot.2022sep23.2324

**Keywords: ** graph, adjacency operator, spectral measure,
dominant vector, dominant vertex

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