Degree Based Topological Coindices Of Some Graphs And Their Chemical Significance

Molecular structure descriptors(or topological indices)are numerical parameters as-sociated with chemical constitution for correlation of chemical structures with their various physical properties,chemical reactivity or biological activity.So far the degree based topological indices are the oldest and most successful class of descriptors.A molecular graph is a representation of the structural formula of a chemical compound whose vertices correspond to the atoms and edges correspond to the chemical bond-s.Topological coindices are among topological indices that consider the non adjacent pairs of vertices of molecular graphs.Throughout this thesis we considered seven well-known topological coindices:the first and second Zagreb coindices,the first and second multiplicative Zagreb coindices,the F-coindex,the third Zagreb coindex and the hyper Zagreb coindex.In this thesis we divided our objectives into three main parts.Firstly,by using graph structural analysis and derivation,we studied the above-mentioned topologi-cal coindices of some molecular graphs that frequently appear in medical,chemical,and material engineering such as benzenoid graphs of different families,graphene sheet and C4C8(S)nanotubes and nanotorus.We obtained the computational formulae of the topological coindices of these important graphs.Furthermore,we analyze the re-sults by MATLAB and obtain the relationship of the coindices which they describe the physcio-chemical properties and biological activities.In addition,the subdivision graph of the Wheel and the line graph of subdivision of the Wheel graph under these topo-logical coindices are discussed where both graphs have various applications in chemical research.Secondly,the Zagreb coindices of product of graphs such as the Cartesian product,disjunction,composition,symmetric difference,tensor product and strong product of two graphs are computed.The first Zagreb coindex of the disjunction,composition and symmetric difference of graphs correct some errors in A.R.Ashrafi et al.(Discrete Applied Mathematics 158(2010)1571-1578).The application of some product graphs are studied and computational formulas are also presented.In addition we introduced a new invariant named as the first general Zagreb coindex which is considered as the generalization of the first Zagreb coindex,the F-coindex and the first general Zagreb index and defined as(?)(G)=?uv?E(G)[dG(u)?+ dG(v)?],where ??R,??0.We do also studied its basic properties and behavior under some graph operations such as union,sum,Cartesian product,composition and tensor product of two or more graphs.As an application,derived results are applied to find chemically important graphs which can be obtained from simpler graphs by applying those studied graph operations.Lastly,the chemical significance and applicability of the above seven topological coindices are studied and their predictive powers are also compared with their corresponding pre-decessor degree based indices.To perform this an important set of organic molecules called octane isomers are considered.These organic compounds are structurally di-verse enough to yield considerable variation in shape,branching and non polarity which makes most suitable to the study.The octane isomers data set consists of 16 physico-chemical properties:the boiling point,melting point,heat capacity at P constant,heat capacity at T constant,Entropy,density,standard enthalpy of vaporization,enthalpy of vaporization,enthalpy of formation,standard enthalpy of formation,motor octane number,molar refraction,acentric factor,total surface area,octanol-water partition coefficient,and molar volume are considered in the study.Finally,concluding remarks along with valuable recommendations axe inferred.