A Review of Application of Graph Theory for Network

In the modern world, planning efficient routes is essential for business and industry, with applications as varied as product distribution. Networks are used to move people, transport goods, communicate information and control the flow of matter and energy. Networks are all around us. Roads, railways, cables, pipelines are phenomena that frequently need to be represented and analyzed as a network. The complexity of network, cost and time required for networking is increasing in different kinds of network based systems. (e.g., T.V. cable networks, Telephone networks, Electricity supply networks, Gas pipe network and water supply system). A graph is a mathematical abstraction that is useful for solving many kinds of problems. Finding shortest paths plays an important role in such kind of network based systems. In graph theory number of algorithms can be applied for finding shortest paths in a graph based network system. It reduces complexity of network paths, cost and time to build and maintain network based systems. In this review of literature authors have reviewed the literatures. This work addresses the problem by presenting analysis of different researches on shortest path problem in various areas of applications. This paper analysis different shortest path algorithms like Dijkstra’s Algorithm, Bellman ford Algorithm and Warshall’s Algorithm by considering network base systems such as Cable network (T.V. cabling, Telephone cabling, Electricity power supply network) and water supply system network. This review of literature also aims to encourage additional research on topics, and concludes with several suggestions for further research. KeywordsGraph, Network based systems, shortest path problem, Dijkstra’s Algorithm, Bellman ford Algorithm and Warshall’s Algorithm. INTRODUCTION A network is a system of points with distances between them. A network can represent roads, pipelines, cables etc. Typical problems with networks involve finding the shortest path between one point in the network and another. In many real occasions, various attributes (various costs and profits) are usually considered in a shortest path problem. Because of the frequent occurrence of such network structured problems, there is a need to develop an efficient procedure for handling these problems. The shortest path algorithm is among a small group of efficient algorithms that exist for this class of problems. In a network, any node may be connected by edges to any number of other nodes. In most representations, a link also has a cost / weight / distance / length that gives the cost for traveling across the link. Finding the shortest paths can give you the best route from one point to another. Think of a map as an instance of a graph, the cities are the nodes and the roads between cities are the edges. The length of the road is the weight of the corresponding edge. Shortest path analysis finds the path with the minimum cumulative impedance between nodes on a network. The path may connect just two nodes – the origin and destination –or have specific stops between the nodes. It is based on a network with the objective of finding the path with the minimum cumulative cost in either time or distance between points on the network. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a directed weighted graph such that the sum of the weights of its constituent edges is minimized. In this study all nodes of graph are represented as places in the city, edges represents roads or paths between places and the weight of the edge represents cost or length of cable or water pipeline. Graph Theory Basics: A graph is a mathematical abstraction that is useful for solving many kinds of problems. Fundamentally, a graph consists of a set of vertices, and a set of edges, where an edge is something that connects two vertices in the graph. A graph is a pair (V,E), where V is a finite set and E is a binary relation on V. V is called a vertex set whose elements are called vertices. E is a collection of edges, where an edge is a pair (u,v) with u,v in V. Graph G = (V, E) is a collection of V nodes connected by E links. This definition of a graph is vague in certain respects; it does not say what a vertex or edge represents. They could be cities with connecting roads, or web-pages with hyperlinks. Path: A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. Undirected Graph: A graph in which each edge symbolizes an unordered, transitive relationship between two nodes. Such edges are rendered as plain lines or arcs. Directed Graph/Digraph: A graph in which each edge symbolizes an ordered, non-transitive relationship between two nodes. Such edges are rendered with an arrowhead at one end of a line or arc. A. B. Sadavare et al, / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 3 (6) , 2012,5296-5300