New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

2013/05/31 by Daniel Lenz, Konstantin Pankrashkin
Mathematics · #Class (philosophy) #Computer science #Discrete mathematics #Graph theory and applications #Laplace operator #Mathematical analysis #Mathematics #Metric (unit) #Operator (biology) #Pure mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.AP #math.SP

paper · pdf · doi:10.1007/s00020-015-2253-2

published as Integral Equations Operator Theory 84 (2016) 151-181 · 32 pages

arxiv created 2013/05/31 · openalex publication_date 2015/07/27 · arxiv updated 2016/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with the solutions of the associated wave equation. The machinery used allows one to study a class of infinite graphs without assumption on the local finiteness.

Citations

• Spectral graph theory [CBMS Conference on Recent Advances in Spectral Graph Theory held at California State University at Fresno, June 6 - 10. 1994]
• SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE SCHRÖDINGER OPERATORS
• Eigenfunction expansion for Schrodinger operators on metric graphs