  Bayesian network (BN) structure discovery algorithms typically either make
assumptions about the sparsity of the true underlying network, or are limited
by computational constraints to networks with a small number of variables.
While these sparsity assumptions can take various forms, frequently the
assumptions focus on an upper bound for the maximum in-degree of the underlying
graph $\nabla_G$. Theorem 2 in Duttweiler et. al. (2023) demonstrates that the
largest eigenvalue of the normalized inverse covariance matrix ($\Omega$) of a
linear BN is a lower bound for $\nabla_G$. Building on this result, this paper
provides the asymptotic properties of, and a debiasing procedure for, the
sample eigenvalues of $\Omega$, leading to a hypothesis test that may be used
to determine if the BN has max in-degree greater than 1. A linear BN structure
discovery workflow is suggested in which the investigator uses this hypothesis
test to aid in selecting an appropriate structure discovery algorithm. The
hypothesis test performance is evaluated through simulations and the workflow
is demonstrated on data from a human psoriasis study.
