, Volume 29, Issue 5, pp 1175-1181
Date: 07 Aug 2012

The Domination Polynomial of a Graph at −1

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Abstract

Let G be a simple graph. The domination polynomial of a graph G of order n is the polynomial \({D(G,x)=\sum_{i=0}^{n} d(G,i) x^{i}}\) , where d(G, i) is the number of dominating sets of G of size i. In this article we investigate the domination polynomial at −1. We give a construction showing that for each odd number n there is a connected graph G with D(G, −1) = n.