  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.

Combinatorics;Spectral Theory