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 ℓ2(V).
Each ξ∈ℓ2(V) defines
a spectral measure μξ on Σ(A);
therefore each v∈V defines the measure μv on Σ(A)
associated with the vector δv∈ℓ2(V).
A vertex v is dominant if, for all w∈V, the measure μw
is absolutely continuous with respect to μv.
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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