Two characteristic polynomials corresponding to graphical networks over min-plus algebra

TL;DR

This paper explores the roots of characteristic polynomials in min-plus algebra, linking them to graph features and introducing new polynomials that reveal additional roots beyond eigenvalues.

Contribution

It introduces new characteristic polynomials for min-plus matrices based on an analogue of the Faddeev-LeVerrier algorithm, capturing multiple roots related to graph structures.

Findings

Minimum roots of the new polynomials match min-plus eigenvalues

Other roots provide additional graph-related information

Illustrative example differentiates between known and new polynomials

Abstract

In this paper, we investigate characteristic polynomials of matrices in min-plus algebra. Eigenvalues of min-plus matrices are known to be the minimum roots of the characteristic polynomials based on tropical determinants which are designed from emulating standard determinants. Moreover, minimum roots of characteristic polynomials have a close relationship to graphs associated with min-plus matrices consisting of vertices and directed edges with weights. The literature has yet to focus on the other roots of min-plus characteristic polynomials. Thus, here we consider how to relate the 2nd, 3rd,... minimum roots of min-plus characteristic polynomials to graphical features. We then define new characteristic polynomials of min-plus matrices by considering an analogue of the Faddeev-LeVerrier algorithm that generates the characteristic polynomials of linear matrices. We conclusively show that minimum roots of the proposed characteristic polynomials coincide with min-plus eigenvalues, and observe the other roots as in the study of the already known characteristic polynomials. We also give an example to illustrate the difference between the already known and proposed characteristic polynomials.