  A matrix-weighted graph is an undirected graph with a $k\times k$ positive
semidefinite matrix assigned to each edge. There are natural generalizations of
the Laplacian and adjacency matrices for such graphs. These matrices can be
used to define and control expansion for matrix-weighted graphs. In particular,
an analogue of the expander mixing lemma and one half of a Cheeger-type
inequality hold for matrix-weighted graphs. A new definition of a
matrix-weighted expander graph suggests the tantalizing possibility of families
of matrix-weighted graphs with better-than-Ramanujan expansion.
