Hafnian of two-parameter matrices

TL;DR

This paper introduces an efficient method for calculating the hafnian of two-parameter matrices, especially for graphs with edges of two weights, with applications in combinatorics and quantum theory.

Contribution

It presents a novel formula for the hafnian of a sum of matrices, enabling exact computation for specific two-parameter matrices like Toeplitz matrices.

Findings

Derived a formula relating hafnians of matrix sums to submatrix hafnians.

Applied the method to count perfect matchings in graphs with two edge weights.

Provided new formulas for counting linear chord diagrams and interpreted integer sequences.

Abstract

The concept of the hafnian first appeared in the works on quantum field theory by E. R. Caianiello. However, it also has an important combinatorial property: the hafnian of the adjacency matrix of an undirected weighted graph is equal to the total sum of the weights of perfect matchings in this graph. In general, the use of the hafnian is limited by the complexity of its computation. In this paper, we present an efficient method for the exact calculation of the hafnian of two-parameter matrices. In terms of graphs, we count the total sum of the weights of perfect matchings in graphs whose edge weights take only two values. This method is based on the formula expressing the hafnian of a sum of two matrices through the product of the hafnians of their submatrices. The necessary condition for the application of this method is the possibility to count the number of k-edge matchings in some graphs. We consider two special cases in detail using a Toeplitz matrix as the two-parameter matrix. As an example, we propose a new interpretation of some of the sequences from the On-Line Encyclopedia of Integer Sequences and then provide new analytical formulas to count the number of certain linear chord diagrams.