# The independence polynomial of inverse commuting graph of dihedral groups

## Authors

• Aliyu Suleiman Bayero University, Kano, Nigeria.
• Aliyu Ibrahim Kiri Bayero University, Kano, Nigeria.

## Keywords:

Dihedral group, inverse commuting graph, independent sets, independence polynomial.

## Abstract

Set of vertices not joined by an edge in a graph is called the independent set of the graph. The independence polynomial of a graph is a polynomial whose coefficient is the number of independent sets in the graph. In this research, we introduce and investigate the inverse commuting graph of dihedral groups (D2N) denoted by GIC. It is a graph whose vertex set consists of the non-central elements of the group and for distinct  x,y, E D2N, x and y are adjacent if and only if xy = yx = 1  where 1 is the identity element. The independence polynomials of the inverse commuting graph for dihedral groups are also computed. A formula for obtaining such polynomials without getting the independent sets is also found, which was used to compute for dihedral groups of order 18 up to 32.

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