Graphs which their certain polynomials have few distinct roots- a survey

TL;DR

This survey explores graphs with specific polynomials that have few roots, revealing surprising structural insights and focusing on domination, independence, and edge cover polynomials.

Contribution

It provides a comprehensive overview of graphs with polynomials having few roots, highlighting their structural properties and implications.

Findings

Graphs with few polynomial roots reveal unique structural characteristics.

Certain classes of graphs exhibit polynomials with limited roots, indicating special properties.

The survey consolidates known results and identifies open problems in this area.

Abstract

Let G = (V;E) be a simple graph. We consider domination polynomial, matching polynomial and edge cover polynomial of G. Graphs which their polynomials have few roots can give sometimes a very surprising information about the structure of the graph. In this paper we study graphs which their domination polynomial, independence polynomial and edge cover polynomial have few roots.