Non-adjacent vertex sum polynomial of Umbrella graphs, Jahangir graph, Tadpole graph and Lollipop graph

Abstract

Let $G=(V, E)$ be a graph. The non-adjacent vertex sum polynomial of the graph $G=(V, E)$ is defined as $N A V S P(G, x)=\sum_{j=0}^{\Delta(G)} n_{(\triangle(G)-j)} x^{\alpha_{\Delta(G)-j}}$ where $n_{\triangle(G)-j}$ is the sum of the number of non-adjacent vertices of all the vertices of degree $\Delta(G)-j$ and $\alpha_{\Delta(G)-j}$ is the sum of the degree of non-adjacent vertices of the vertices of degree $\Delta(G)-j$. In this paper we derived the non-adjacent vertex sum polynomial for Umbrella graph, Jahangir graph, Tadpole graph and Lollipop graph.