Abstract

The average domination polynomial of graphs is unimodal

Highlights

  • Throughout this paper, we consider only simple graphs

  • The domination polynomial of G that is denoted by D(G, x) is the one variable polynomial such that the coefficient of xk is dk, where dk is the number of dominating sets of G with size k

  • It is known that any log-concave polynomial with positive coefficients is unimodal

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Summary

Introduction

Throughout this paper, we consider only simple graphs (the graphs with no loops and multiple edges). For every n be the average of the domination polynomials of all labeled graphs on n vertices. The polynomial Φn(x) is studied and it is shown that Φn(x) is log-concave and unimodal. By studying these polynomials one can obtain some properties of a graph. The domination polynomial of G that is denoted by D(G, x) is the one variable polynomial such that the coefficient of xk is dk, where dk is the number of dominating sets of G with size k.

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