Domination polynomial of generalized friendship and generalized book graphs

TL;DR

This paper investigates the domination polynomials of generalized friendship and book graphs, analyzing their roots and limiting behaviors to deepen understanding of their combinatorial properties.

Contribution

It introduces the domination polynomials for these graph families and explores their roots and asymptotic behavior, which was not previously studied.

Findings

Derived explicit formulas for domination polynomials of the graphs.

Identified the distribution and limiting curves of domination roots.

Provided insights into the asymptotic behavior of domination roots.

Abstract

Let G be a simple graph of order n. The domination polynomial of a graph is the generating function of its dominating sets. We study the domination polynomials of generalized friendship graphs. We also consider book graphs formed by joining n copies of the cycle graph of order 4 with a common edge and study the domination polynomials of some generalized book graphs. In particular we examine the domination roots of these families and find the limiting curve for the roots.