# There is no going back: Properties of the non-backtracking Laplacian

## TL;DR

This paper explores the properties of the non-backtracking Laplacian and graph, establishing isomorphism criteria, analyzing eigenfunctions, introducing circularly partite graphs, and deriving spectral bounds.

## Contribution

It introduces circularly partite graphs, proves graph isomorphism via non-backtracking graphs, and analyzes eigenfunctions and spectral bounds of the non-backtracking Laplacian.

## Key findings

- Two graphs are isomorphic iff their non-backtracking graphs are isomorphic.
- Sharp upper bounds for spectral gap from 1 are established.
- Relations between singular values of the Laplacian and independence numbers are shown.

## Abstract

We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking graphs are isomorphic, and we investigate properties of various classes of non-backtracking Laplacian eigenfunctions, such as symmetric and antisymmetric eigenfunctions. Moreover, we introduce and study circularly partite graphs as a generalization of bipartite graphs, and we use this notion to state a sharp upper bound for the spectral gap from $1$. We also investigate the singular values of the non-backtracking Laplacian in relation to independence numbers, and we use them to bound the moduli of the eigenvalues.