In Electrical Engineering , graph theory is used in designing of circuit connections. These circuit connections are named as topologies. Some topologies are series, bridge, star and parallel topologies. In mathematics, operational research is the important field. Graph theory provides many useful applications in operational research. Graphs are used to represent the routes between the cities. With the help of tree that is a type of graph, we can create hierarchical ordered information such as family tree.

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A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another 1. Euler becomes the father of graph theory as well as Topology.

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The paper written by Leonhard Euler on the Seven Bridges of Konigsberg and published in is regarded as the first paper in the history of graph theory. This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L'Huilier, and represents the beginning of the branch of mathematics known as topology. This study had many implications for theoretical chemistry. Now we have the following graph by considering these sets.

Every graph as a diagram associated with it. If two distinct edges say e1 and e2 are incident with common vertex, then they are adjacent edges. A graph with p-vertices and q-edges is called a p,q graph. The 1, 0 graph is called trivial graph. In Directed Graph each edge is represented by an arrow or direction curve from initial point u of e to the terminal point v fig.

Otherwise infinite graph.

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In other words, isolated vertices are those with zero degree. A vertex of degree one is called Pendent vertex or End vertex. Statement: The number of vertices of odd degree in a graph is always even Problem 2. Determine the number of edges in a graph with 6 vertices, 2 of degree 4 and 4 of degree 2. Draw two such graphs. Suppose the graph with 6 vertices has e number of edges.

## Electronic Journal of Graph Theory and Applications

Two such graphs are shown below in figure 7. Null graph is denoted on n vertices by Nn N4 is shown in the finger 8. A complete graph is usually denoted by Kn. It should be noted that Kn has exactly edges. If the degree of each vertex is r, then the graph is called a regular graph of degree r.

The wheels W3, W4, W5 and W6 are The adjacent if and only if the bit strings that they represent differ in exactly one bit position. The graphs Q1 ,Q2 ,Q3 are displayed in the figure Thus Qnhas 2n vertices and n. An internal vertex or inner vertex or branch vertex is a vertex of degree at least 2.

Similarly, an external vertex or outer vertex, terminal vertex or leaf is a vertex of degree 1. An irreducible tree or series-reduced tree is a tree in which there is no vertex of degree 2. Equivalently, a forest is an undirected acyclic graph. As special cases, an empty graph, a single tree, and the discrete graph on a set of vertices that is, the graph with these vertices that has no edges , are examples of forests.

In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A directed tree is a directed graph which would be a tree if the directions on the edges were ignored, i. Some authors restrict the phrase to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex see arborescence 4.

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More formally, a tree is starlike if it has exactly one vertex of degree greater than 2. In other words, a star tree of order n is a tree of order n with as many leaves as possible.

Therefore G and G' are said to be isomorphic. The following graphs are isomorphic to each other. The following two graphs are not isomorphic, because x is adjacent to two pendent vertex is not preserved. The below image shows a union of graph G and graph H. Make sure that all the vertices and edges from both the graphs are present in the union. If possible draw it once yourself. It is analogous to the disjoint union of sets, and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the result be the disjoint union of the edge sets of the given graphs.

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Any disjoint union of two or more nonempty graphs is necessarily disconnected. Below image demonstrate the Intersection of the two Graphs.

Notice that the vertex set is a union of both the vertex sets but the edge family consists only the edges which exist in both the graphs G and H. Below is an image to demonstrate that. Below is a demonstration of Graph Complement. Which means we will always say that the kth power of G and not just power of G. The kth power of G is a graph with the same set of vertices as G and an edge between two vertices in the Power Graph exists only if there is a path of length at most k between them. So the graph will result into the graph on the right. Note that the vertices 1 and 3 are connected because the there exists a path of length two or less in the original graph..

This concludes that all the graph are the first power of themselves. Which means if value of k is 1 then we get the same graph? And if the adjacency matrix of G is A then each cell i,j of A1 will give you the number of paths of length 1 between i and j. Lets take another example. Again let the value of k is 2. Consider the graph below. No edge appears more than once. It is also called as an edge train or a chain. That is, a circuit is a closed trail.

The number of edges in the path is called length of a path. Otherwise, G is disconnected. Each of these connected subgraph is called a component. The matrix are commonly used to represent graphs for computer processing. The advantages of representing the graphs in matrix form lies on the fact that many resut of matrix algebra can be readily applied to study the structures of graphs from an algebraic point of view. Two types representation are given below: Node: Inter connection of components.

Branch: Line connecting two nodes. Circuit Here a, b, c and d are called Nodes and ab, bc, cd, db, and ad are Branches. Here we take the values as per the direction. We will take those branches which leads to loops in graph. W e consider the cut set branches 1, 3, 4 and 7. If not we take -1 at which the node at the head of the branch is considered and takes zero if the branch is not connected to that node. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B Can be also represented as directed graph.

In general graphs theory has a wide range of applications in diverse fields. This paper explores different elements involved in graph theory including graph representations using computer systems and graph-theoretic data structures such as list structure and matrix structure. The emphasis of this paper is on graph applications in computer science. To demonstrate the importance of graph theory in computer science, this article addresses most common applications for graph theory in computer science. These applications are presented especially to project the idea of graph theory and to demonstrate its importance in computer science.

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