We consider score tests of the null hypothesis ${\rm H}_0: \theta = \frac{1}{2}$ against the alternative hypothesis ${\rm H}_1: 0 \leq \theta < \frac{1}{2}$, based upon counts multinomially distributed with parameters $n$ and $\rho(\theta,\pi)_{ 1 \times m} = \pi_{1 \times m} T(\theta)_{m \times m}$, where $T(\theta)$ is a transition matrix with $T(0) = I$, the identity matrix, and $T(\frac{1}{2}) = {\bf 1}^T \alpha$, ${\bf 1} = (1,\ldots, 1)$. This type of testing problem arises in human genetics when testing the null hypothesis of no linkage between a marker and a disease susceptibility gene, using identity by descent data from families with affected members. In important cases in this genetic context, the score test is independent of the nuisance parameter $\pi$ and is based on a widely used test statistic in linkage analysis. The proof of this result involves embedding the states of the multinomial distribution into a continuous time Markov chain with infinitesimal generator $Q$. The second largest eigenvalue of $Q$ and its multiplicity are key in determining the form of the score statistic. We relate $Q$ to the adjacency matrix of a quotient graph, in order to derive its eigenvalues and eigenvectors.