Research On The Three Polynomials And Topological Indices Of Some Special Graphs

In this thesis,we let that G =(V,E)be a simple graph of order |V|= n.A set S C V is called to be a dominating set of G if every vertex in V/S is adjacent to at least one vertex in S.The domination polynomialis the ordinary generating function for the number of dominating sets in a graph G,where d(G,i)is the number of dominating sets of G of cardinality i,and ?(G)is the domination number of G.A vertex subset S(?)V of the graph G is an independent dominating set if every vertex in V/S is adjacent to at least one vertex in S and the vertices of S are pair wise non-adjacent.The independent domination polynomial is the ordinary generating function for the number of independent dominating sets in a graph G,where di(G,j)is the number of independent dominating sets of size j,and?i(G)is the cardinality of a smallest independent dominating set of a graph G.A set S(?)V is called a total dominating set if every vertex of G is adjacent to some vertex in S.The cardinality of a smallest total dominating set is called total domination number of a graph G,and it denoted by ?t(G).The total domination polynomial is the ordinary generating function for the number of total dominating sets in a graph G,where dt(G,i)is the number of total dominating sets of G of cardinality i.We are divided our objectives in this thesis into two parts.In the first part we investigated and studied the domination polynomial,the independent domination polynomial,and the total domination polynomial and their related terminologies for some special graphs.So we studied and investigated these three polynomials in more details,and obtained some of their properties.Furthermore,the properties of their coefficients are as well obtained.Also we computed and derived new formulas of these polynomials for some special graphs.These polynomials have many useful properties in the graph theory.In the second part we studied the molecular topological properties of some special graphs.The indices atom-bond connectivity(ABC)index,fourth version(ABC4)of atom-bond connectivity index,geometric-arithmetic(GA)index and fifth version of geometric-arithmetic(GA5)of these special graphs are computed.