  We consider graphs for which the non-backtracking matrix has defective
eigenvalues, or graphs for which the matrix does not have a full set of
eigenvectors. The existence of these values results in Jordan blocks of size
greater than one, which we call nontrivial. We show a relationship between the
eigenspaces of the non-backtracking matrix and the eigenspaces of a smaller
matrix, completely classifying their differences among graphs with at most one
cycle. Finally, we provide several constructions of infinite graph families
that have nontrivial Jordan blocks for both this smaller matrix and the
non-backtracking matrix.
