  This paper aims to determine the ring structure of the torus equivariant
cohomology of odd-dimensional complex quadrics by computing the graph
equivariant cohomology of their corresponding GKM graphs. We show that its
graph equivariant cohomology is generated by three types of subgraphs in the
GKM graph, which are subject to four different types of relations. Furthermore,
we consider the relationship between the two graph equivariant cohomology rings
induced by odd- and even-dimensional complex quadrics.
