The Applications Of Graph Theory In Communication Networks And The Research On Molecular Topological Indices

Information communication network is one of important infrastructure of modern information society.The people's service requirements for information networks promote rapid development of communication technologies and networks.In the communication networks,by establishing mathematical models of graph theory,network planning and flow optimization are one of important methods to improve and ensure the quality of network communication.As a rapidly developing research tool in combinatorial optimization,graph theory has been widely used to computate the time of information transmission in communication systems.Molecular topological index is an important mathematical descriptor of molecular structure in the research of quantitative structure property/activity relations(QSPR/QSAR)of compounds.Topological index is closely related to the physicochemical properties and biological activities of the corresponding compounds.Furthermore,topological index can be used to predict the physicochemical properties of new compounds.The main contents of this paper are as follows:(1)We introduced the research background and significance,basic concept of this paper and the development and application of graph theory.(2)The basic concepts of Scrambling index and generalized Competition index were introduced,and introduced its applications in communication systems.We gave the upper bound for the Scrambling index of primitive ?-graphs and primitive extended ?-graphs,and obatined some extremal graphs.The generalized Competition indices of some class of extremal digraphs in Scrambling indices of primitive digraphs were obtained.The above results can be used to computate the minimum time of information transmission in communication systems.Using software of Matlab,some examples were given to illustrate our results.(3)The research background of augmented Zagreb index was introduced.The augmented Zagreb index of catacondensed hexagonal systems was provided and the catacondensed hexagonal systems with minimal and maximal augmented Zagreb indices were determined.The lower bound for augmented Zagreb index of molecular trees with fixed numbers of pendant vertices was obtained,and the extremal trees were characterized.(4)The harmonic index was studied.A upper bound and a lower bound on the sum of harmonic index and average eccentricity(the product of harmonic index and average eccentricity,respectively)were presented,and the corresponding extremal graphs were characterized.A upper bound on the harmonic index for chemical graphs with k pendant vertices was provided,and the extremal graphs were determined.The sharp lower and upper bounds on the harmonic index of quasi-tree graphs were given and the corresponding extremal graphs were characterized.