# On the Characteristic and Permanent Polynomials of a Matrix

## TL;DR

This paper introduces a recurrence relation approach to compute the characteristic and permanent polynomials of a matrix using its associated digraph, revealing a combinatorial interpretation involving vertex-disjoint subdigraphs.

## Contribution

It presents a novel recurrence-based method to compute characteristic and permanent polynomials via induced subdigraphs, offering a combinatorial perspective.

## Key findings

- Characteristic and permanent polynomials can be computed from subdigraphs.
- Induced subdigraphs are vertex-disjoint and partition the original digraph.
- Provides a combinatorial interpretation of matrix invariants.

## Abstract

There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic, and permanent polynomials can be calculated in terms of characteristic, and permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic, and permanent polynomials; the determinant, and permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.