Domination Forgotten and Wiener Polynomials of Certain Graphs

Abstract

Graph Theory is a newly emerging discipline in mathematics that has significant applications. Domination is an area in Graph Theory with an extensive research activity. Let G = (V,E) be a simple graph. A set S and#8838; V is a dominating set of G, if every vertex in newline(V-S) is adjacent to atleast one vertex in S. We construct Fk the family of all dominating newlinesets of a Fan graph Fm with cardinality k. Let d(Fm, k) =| Fk |. After construction of newlinek obtain a recursive formula, for d(Fm, k). Using this recursive formula, consider the newlinepolynomial d(Fm, x) = and#931;m=and#947;(Fm)d(Fm, k)xk the domination polynomial of Fan graph Fm Several properties and results of domination sets and domination polynomial of Fan graph Fm are proved. Double Fan graph Dm and Triangular Book graph B3,n are also discussed. newlineA chemical graph can be recognized by a numerical number (topological index), algebraic polynomial or any matrix. These numbers and polynomial help to predict newlinemany physic-chemical properties of underline chemical compound. We compute forgotten polynomial of some graphs, subdivision graphs, nCnPk+1, CkPu+1 Silicon Carbide types, newlineEnhanced Mesh and Triangular Mesh. newlineThe Wiener index is a graphical invariant that has found extensive application in Chemistry. Wiener index is the first reported distance based topological index defined as half sum of the distances between all the pair of vertices in a molecular graph. We find Wiener polynomial of Triangular Snake graph Tn, Butterfly graph Bfm,n and DutchWindmill graphs and study its properties. newline newline newline