# Norms, Kernels and Eigenvalues of some Infinite Graphs

## TL;DR

This paper investigates the spectral properties of adjacency matrices of certain infinite graphs, providing norm estimates, kernel triviality conditions, and eigenvalues for specific graph constructions.

## Contribution

It introduces new methods for analyzing the norms, kernels, and eigenvalues of infinite graphs, especially those formed by attaching infinite tails to finite graphs.

## Key findings

- Established norm estimates for the adjacency operator
- Determined the triviality of kernels for some infinite trees
- Computed eigenvalues for graphs with infinite tails

## Abstract

In these paper we study the adjacency matrix of some infinite graphs, which we call the shift operator on the $L^p$ space of the graph. In particular, we establish norm estimates, we find the norm for some cases, we decide the triviality of the kernel of some infinite trees, and we find the eigenvalues of certain infinite graphs obtained by attaching an infinite tail to some finite graphs.