Molecular biology is a branch of biology that deals with the molecular basis of biological activity . Molecular biology chiefly concerns itself with understanding the interactions between the various systems of a cell , including the interactions between the different types of DNA , RNA and protein biosynthesis as well as learning how these interactions are regulated . In recent years several spectacular events connected with genome studies have occurred , reading the human genome being one of them . Since a discovery by Watson and Crick their double helix model of a DNA chain , biology has made a great progress in nderstanding foundations of life . The progress would have not been possible , however , without a help from other areas of science , especially its part connected with mathematics . It is desirable to explore mathematical tools for efficient extraction of information from such sources . It is worth highlighting that graph theory is a fine instance of pure mathematics that has found a variety of applications in the course of time . Created in the works of L . Euler (1707–1783) in the eighteenth century . The principles of graph theory , which was earlier applied in fields such as electrical engineering and computer networks are now being adopted to investigate protein structure , DNA sequencing and other problem in molecular biology . In this thesis , we present a survey on using of graph theoretical techniques in molecular biology in the first part and in the second part , we study the application of molecular biology in solving the hard problem in graph theory . The organization of the thesis is as follows . In the first chapter , we introduce a few basic concepts of graph theory and molecular biology that are necessary to understand the subsequent exposition . In Part 1 , Chapter 2 discusses the DNA sequencing problem and the DNA assembling and application of graph theory in solving them are discussed . The graph theory approach to RNA structures has implications for RNA genomics , structure analysis and design that are disscussed in Chapter 3 . In Chapter 4 , we summarize current applications and development of graph theory modeling in protein identification , and the manner in which graphs are analyzed and parameters relevant to protein structure are extracted , are explained . The structural and biological information derived from protein structures using these methods is presented . In Chapters 5 and 6 , we discuss recent work on identifying and modelling the structure of bio-molecular networks and applying graph theory to drug discovery and design with use of some important topological indices . In Part 2 , we will concentrate on the approach for solving hard problem especially hard problem in graph theory by molecular biology called DNA computing . DNA computing is a form of computing which uses DNA , biochemistry and molecular biology , instead of the traditional silicon-based computer technologies . DNA computing , or more generally , biomolecular computing , is a fast developing interdisciplinary area . Research and development in this area concerns theory , experiments , and applications of DNA computing.\\ This field was initially developed by Leonard Adleman of the University of Southern California , in 1994 . Adleman demonstrated a proof-of-concept use of DNA as a form of computation which solved the seven-point Hamiltonian path problem . Since the initial Adleman experiments , advances have been made and various Turing machine have been proven to be constructible . In Chapter 7 , we discuss algorithms that use of different thechniques of DNA computing for solving some important NP problem of graph theory in polynomial number of biological operations .