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An understanding of the vocabulary terms is critical for successful lesson completion. Then have a discussion about these terms and their meanings so that everyone understands them. In this lesson, you will explore the concepts of graph theory. Many different types of engineers represent different situations using graph theory.

Electrical engineers and computer engineers use graph theory to represent integrated circuits. By using graph theory components, density can be maximized to optimize processing speed and electrical efficiency. Network engineers use graph theory to represent communication networks with terminals and relay stations as the nodes.

Communication links between the network devices are the edges. Any situation that has linked items can be represented using graph theory.

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The nodes represent the pinned joints in the truss. Edges represent the rods in the truss with its end vertices corresponding to the joints that connect the rod to the truss. Edges are also used to represent external forces and reactions. Can you think of any possible applications for graph theory?

Believe it or not, some engineers use graph theory to understand relationships between people.

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This is the case for engineers who work for companies such as Facebook, where an understanding of relationships between people is critical for advertisement and revenue. In this lesson, and its associated activities, we will focus on using graph theory to represent relationships between people. At the same time, we want to remember that many other engineering applications use graph theory. Questioning will be used to guide you to think about graph structures, purposes and layouts.

As you answer the following questions, you will write and draw graphics. A graphic representation of relationships. As the questioning progresses, expect students to begin to see real-world implications of graph theory and how it could be used to represent existing relationships.

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Then, students can use the associated activities, Graphing Your Social Network and Using Graph Theory to Analyze Drama , to practice applying this technique to real world examples. Figure 1. An example vertex and edge graph. Graph theory is the mathematical study of graphs. A graph consists of a set or collection of vertices or nodes and a set or collection of edges that connect the vertices. Vertices that are connected by an edge are said to be adjacent. The degree of a vertex is the number of edges connecting to the vertex as an end point.

In the Figure 1 example graph, all vertices have a degree of 2. Some of the different types of graphs are the complete graph and the regular graph. A complete graph is a graph in which every vertex is adjacent to every other vertex. A regular graph has the same degree on all vertices. Figure 2. Adjacency matrix for the Figure 1 graph. An adjacency matrix for a graph with n vertices is an n x n matrix where entry i, j is 1 if there is an edge in the graph between vertex i and j, otherwise the entry is 0.

Research Topics in Graph Theory and Its Applications

The adjacency matrix shown in Figure 2 is for the graph shown in Figure 1. An adjacency list for a graph has a row for each vertex and contains the vertex label followed by the vertex labels of adjacent vertices. The adjacency list for the graph in Figure 1 is shown in Figure 3. Figure 3.

Adjacency list for the Figure 1 graph. A path in a graph from vertex v to u is a sequence of edges where the first edge starts at v, each successive edge connects to the previous edge, and the final edge terminates at u. A path from 1 to 2 in the Figure 1 graph would be: 1, 3 , 3, 2. A vertex v is reachable from vertex u, if there is a path from v to u.

Graph theory, combinatorics, and their applications to engineering and information sciences

A graph is connected , if every vertex is reachable from every other vertex, otherwise the graph is considered disconnected. Sibling vertices are both children to the same parent. Observations: As students are engaged in the lesson, ask the following or similar questions:.

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  • Writing: Assign students to complete the following writing prompts, as provided on the Graph Theory in Drama Summative Assessment , and below. Review their answers to assess their mastery of the subject matter before conducting the associated activities. It is recommended to conduct the social networks activity Graphing Your Social Network first and then the dramatic networks activity Using Graph Theory to Analyze Drama , but the order is not that critical. In both activities, students gain practice with graph theory vocabulary, including node, edge, betweeness centrality and degree on interaction, and learn about a range of engineering applications of graph theory.

    However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government. Why K engineering? Find more at TeachEngineering. Quick Look. Grade Level: 11 Time Required: 30 minutes Lesson Dependency Lesson dependency indicates that this lesson relies upon the contents of the TeachEngineering document s listed. In the proposed conference, we have planned to conduct a symposium on graph decompositions.

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    Graph decompositions have a wide variety of applications such as computer architecture, designing reliable communication networks. Hypercubes, Butterfly network and de Bruijn network are proved to be good interconnection networks. Graph Designs are used to construct codes which have been used to detect errors in communications. Modern applications are also found in a wide gamut of areas including; mathematical biology, networking, cryptography and medicine.

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    Last Date for abstract submission : October 10, Last Date for full paper submission : October 20, Last Date for Registration : October 30, Confirmation of Registration : November 05, Participants may avail rooms in the college hostel. Each room will be shared by 2 or 3 participants. Details about the room allotment will be intimated around November 10 and will be updated on the website as well.

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