  We define a novel notion of ``non-backtracking'' matrix associated to any
symmetric matrix, and we prove a ``Ihara-Bass'' type formula for it.
  We use this theory to prove new results on polynomial-time strong refutations
of random constraint satisfaction problems with $k$ variables per constraints
(k-CSPs). For a random k-CSP instance constructed out of a constraint that is
satisfied by a $p$ fraction of assignments, if the instance contains $n$
variables and $n^{k/2} / \epsilon^2$ constraints, we can efficiently compute a
certificate that the optimum satisfies at most a $p+O_k(\epsilon)$ fraction of
constraints.
  Previously, this was known for even $k$, but for odd $k$ one needed $n^{k/2}
(\log n)^{O(1)} / \epsilon^2$ random constraints to achieve the same
conclusion.
  Although the improvement is only polylogarithmic, it overcomes a significant
barrier to these types of results. Strong refutation results based on current
approaches construct a certificate that a certain matrix associated to the
k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type
argument, from an application of Grothendieck's inequality, or from a spectral
bound obtained with a trace argument. The first two approaches require a union
bound that cannot work when the number of constraints is $o(n^{\lceil k/2
\rceil})$ and the third one cannot work when the number of constraints is
$o(n^{k/2} \sqrt{\log n})$.
  We further apply our techniques to obtain a new PTAS finding assignments for
$k$-CSP instances with $n^{k/2} / \epsilon^2$ constraints in the semi-random
settings where the constraints are random, but the sign patterns are
adversarial.
