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We prove new properties of the non-backtracking graph and thenon-backtracking Laplacian for graphs. In particular, among other results, weprove that two simple graphs are isomorphic if and only if their correspondingnon-backtracking graphs are isomorphic, and we investigate properties ofvarious classes of non-backtracking Laplacian eigenfunctions, such as symmetricand antisymmetric eigenfunctions. Moreover, we introduce and study circularlypartite graphs as a generalization of bipartite graphs, and we use this notionto state a sharp upper bound for the spectral gap from $1$. We also investigatethe singular values of the non-backtracking Laplacian in relation toindependence numbers, and we use them to bound the moduli of the eigenvalues.

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