Equivariant cohomology of odd-dimensional complex quadrics from a combinatorial point of view

This paper aims to determine the ring structure of the torus-equivariant cohomology with integer coefficients of odd-dimensional complex quadrics by computing the graph equivariant cohomology of their corresponding GKM graphs. We show that this graph equivariant cohomology is generated by three types of subgraphs in the GKM graph, which satisfy four distinct types of relations. Furthermore, we investigate the relationship between the graph equivariant cohomology rings associated with odd- and even-dimensional complex quadrics.