  The conjugation action of the complex orthogonal group on the polynomial
functions on $n \times n$ matrices gives rise to a graded algebra of invariant
polynomials. A spanning set of this algebra is in bijective correspondence to a
set of unlabeled, cyclic graphs with directed edges equivalent under dihedral
symmetries. When the degree of the invariants is $n+1$, we show that the
dimension of the space of relations between the invariants grows linearly in
$n$. Furthermore, we present two methods to obtain a basis of the space of
relations. First, we construct a basis using an idempotent of the group algebra
referred to as Young symmetrizers, but this quickly becomes computationally
expensive as $n$ increases. Thus, we propose a more computationally efficient
method for this problem by repeatedly generating random matrices using a Monte
Carlo algorithm.
