We consider score tests of the null hypothesis $(\rm H0: \theta = \frac(1)(2)$ 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.