Smith Normal Form in Combinatorics

This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.Comment: 17 pages, 3 figure

On Smith normal forms of qq-Varchenko matrices

In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the hyperplane arrangement for the regular $n$-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the $q$-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over $\mathbb Z[q]$ and realize their congruent transformation matrices over $\mathbb Z[q]$ as well.Comment: 22 pages, 8 figure

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 concerned on the invariant factors of these q-matrices. In this paper, we put this problem to a graph theory model. We will discuss some general properties and give some methods for finding the invariant factors of q-matrices of some 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. Key words and phrases : q-matrix, invariant factors, bipartite graph, hyperplane arrangement. AMS 1991 subject classification : 05C30, 05C12 1. Introduction and background Let H = fH 1 ; : : : ; H t g be a set of hyperplanes in R n . It is called an arrangement (or a configuration) of hyperplanes. Let r(H) = fR 1 ; : : : ; Rm g be the sets of regions in the complement of the ..

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 bi-partite graphs, even cycles, some hexagonal systems, and some polygonal trees are found. Key words and phrases: q-matrix, invariant factors, bipartite graph, hyperplane arrangement. AMS 1991 subject classification: 05C30, 05C12 1. Introduction an