  Motivated by community detection, we characterise the spectrum of the
non-backtracking matrix $B$ in the Degree-Corrected Stochastic Block Model.
  Specifically, we consider a random graph on $n$ vertices partitioned into two
equal-sized clusters. The vertices have i.i.d. weights $\{ \phi_u \}_{u=1}^n$
with second moment $\Phi^{(2)}$. The intra-cluster connection probability for
vertices $u$ and $v$ is $\frac{\phi_u \phi_v}{n}a$ and the inter-cluster
connection probability is $\frac{\phi_u \phi_v}{n}b$.
  We show that with high probability, the following holds: The leading
eigenvalue of the non-backtracking matrix $B$ is asymptotic to $\rho =
\frac{a+b}{2} \Phi^{(2)}$. The second eigenvalue is asymptotic to $\mu_2 =
\frac{a-b}{2} \Phi^{(2)}$ when $\mu_2^2 > \rho$, but asymptotically bounded by
$\sqrt{\rho}$ when $\mu_2^2 \leq \rho$. All the remaining eigenvalues are
asymptotically bounded by $\sqrt{\rho}$. As a result, a clustering
positively-correlated with the true communities can be obtained based on the
second eigenvector of $B$ in the regime where $\mu_2^2 > \rho.$
  In a previous work we obtained that detection is impossible when $\mu_2^2 <
\rho,$ meaning that there occurs a phase-transition in the sparse regime of the
Degree-Corrected Stochastic Block Model.
  As a corollary, we obtain that Degree-Corrected Erd\H{o}s-R\'enyi graphs
asymptotically satisfy the graph Riemann hypothesis, a quasi-Ramanujan
property.
  A by-product of our proof is a weak law of large numbers for
local-functionals on Degree-Corrected Stochastic Block Models, which could be
of independent interest.
