Adjacent vertex sum polynomial for the splitting graph of Factographs

Abstract

Let $G=(V, E)$ be a graph. The vertex polynomial of the graph $G=(V,E) $ is defined as $V(G,x)=\sum\limits_{k=0}^{\Delta(G)}{v_{k}x^{k}}$, where $\Delta (G)=\max\{d(v)/v\in V\}$ and $v_{k}$ is the number of vertices of degree $k$. The adjacent vertex sum polynomial is defined as $S(G, x)=\sum\limits_{i=o}^{\Delta(G)}{n_{\Delta(G)-i}x^{\alpha_{\Delta (G)-i}}}$, where $n_{\Delta(G)-i}$ is the sum of the number of adjacent vertices of all the vertices of degree $\Delta(G)-i$ and $\alpha_{\Delta(G)-i}$ is the sum of the degree of adjacent vertices of all the vertices of degree $\Delta(G)-i$. In this paper we seek to find the vertex polynomial and the adjacent vertex sum polynomial for the splitting graph of Perfect factograph and the splitting graph of Integral Perfect factograph.