Permanents of Circulants: a Transfer Matrix Approach (Expanded Version)

TL;DR

This paper introduces a combinatorial transfer matrix approach to compute the permanent of circulant matrices, extending previous results and enabling the counting of various structures in circulant graphs.

Contribution

It presents a new combinatorial method using transfer matrices to compute permanents of circulant matrices, extending to non-fixed jumps and counting other graph structures.

Findings

Reproves Minc's polynomial-time result for fixed-jump circulants using combinatorics.

Shows transfer matrices have smaller characteristic polynomials than expected.

Extends permanent calculation to circulants with linear jumps and counts Hamiltonian cycles.

Abstract

Calculating the permanent of a (0,1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0,1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cycle-covers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected. An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps. It also permits us to count other types of structures in circulant graphs, e.g., Hamiltonian Cycles.