  We investigate multipartite information and entanglement measures in the
ground state of a free-fermion model defined on a Hamming graph. Using the
known diagonalization of the adjacency matrix, we solve the model and construct
the ground-state correlation matrix. Moreover, we find all the eigenvalues of
the chopped correlation matrix when the subsystem consists of $n$ disjoint
Hamming subgraphs embedded in a larger one. These results allow us to find an
exact formula for the entanglement entropy of disjoint graphs, as well as for
the mutual and tripartite information. We use the exact formulas for these
measures to extract their asymptotic behavior in two distinct thermodynamic
limits, and find excellent match with the numerical calculations. In
particular, we find that the entanglement entropy admits a logarithmic
violation of the area law which decreases the amount of entanglement compared
to the area law scaling.
