Invariant factors of graphs associated with hyperplane arrangements

A matrix called Varchenko matrix associated with a hyperplane arrangement was defined by Varchenko in 1991. Matrices that we shall call q-matrices are induced from Varchenko matrices. Many researchers are interested in the invariant factors of these q-matrices. In this paper, we associate this problem with a graph theoretic model. We will discuss some general properties and give some methods for finding the invariant factors of q-matrices of certain types of graphs. The proofs are elementary. The invariant factors of complete graphs, complete bipartite graphs, even cycles, some hexagonal systems, and some polygonal trees are found.