Derivatives and real roots of graph polynomials

TL;DR

This paper surveys the properties of derivatives and real roots of graph polynomials, highlighting their applications across various fields and introducing polynomial reconstruction methods for matching polynomials.

Contribution

It provides a comprehensive overview of recent results on derivatives and real roots of graph polynomials and introduces polynomial reconstruction techniques for matching polynomials.

Findings

Analysis of derivatives of graph polynomials

Characterization of real roots in various applications

Introduction of polynomial reconstruction methods

Abstract

Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the derivative and real roots of graph polynomials, which have applications in chemistry, control theory and computer science. Related to the derivatives of graph polynomials, polynomial reconstruction of the matching polynomial is also introduced.