# Talk by Davide Bianchi

On April 27th, 2022, Dr. Davide Bianchi (Harbin Institute of Technology, Shenzen) gave a talk about **"Asymptotic spectra of large graphs with a uniform local structure" **as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

We are concerned with sequences of graphs with a uniform local structure. The underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown that we can associate to it a symbol f, [2]. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution.

We discuss different applications and provide numerical examples in order to underline the practical use of the developed theory, [1]. In particular, we show how the knowledge of the symbol f

• can benefit from iterative methods to solve Poisson equations on large graphs;

• provides insight on the recurrence/transience property of random walks on graphs.

References

[1] A. Adriani, D. Bianchi, P. Ferrari, and S. Serra-Capizzano. Asymptotic spectra of large (grid) graphs with a uniform local structure (part II): numerical applications. 2021. arXiv:2111.13859.

[2] A. Adriani, D. Bianchi, and S. Serra-Capizzano. “Asymptotic spectra of large (grid) graphs with a uniform local structure (part I): theory”. In: Milan Journal of Mathematics 88 (2020), pp. 409–454.