Application of Connectivity in Electric Circuits Submitted to: Mr. Harris R. Dela Cruz Mathematics Department College of Science Bulacan State University in partial fulfillment of the requirements for the course MAT306 - Graph Theory with Applications of the program BS Mathematics w/ specialization in Computer Science. by Earlomatic - BSM CS-3B Ariella S. Diminsil Angelo C. Gutierrez Earl Patrick F. Paguio 2nd Semester, AY 2020-2021 Application of Connectivity in Electric Circuits Ariella S. Diminsil, Angelo C. Gutierrez, Earl Patrick F. Paguio Abstract The goal of this study is to gain a better understanding of the optimization of the electric circuit, the application of graph theory and be able to formulate ideas about the possible electric circuits of the house and its connectivity using graphs. An electric circuit allows a person to utilize the electrical energy that is running in the wires of your home or business. The Application of Connectivity in Electric Circuits is a study that might assist the prospective homeowner in avoiding power overloading and other circuit issues. 1 Introduction Electrical circuits are the fundamental components of electrical appliances. Today's civilization would be completely paralyzed without the use of electrical appliances. Graph theory, a branch of mathematics, now has an impact on almost every other branch of mathematics. The flow of electricity is controlled by an electrical circuit. Electricity is one of the needs inside a household, having it at home makes life easier, every house and buildings have their own unique electronic circuit design. One of the steps in building a house is rough electrical. We will use graph theory to measure the connectivity of every wire inside a house. We decided to observe and discover what must be the best possible electric circuit in the house of the Diminsil Family since they decided to continue the construction of their house located in Macabebe, Pampanga. The first floor of the house is already finished, while the second floor is still ongoing. 2 Preliminary To better understand the optimization of the electric circuit and the application of graph theory, we must know the kits that we will use to obtain our goal. In this section, we will be able to gain knowledge and formulate ideas on how we will determine the possible electric circuit of the house, how graphs help us simplify the problem, and show us the connectivity that we are aiming to solve. 2.1. Graph Graph Theory is ultimately the study of relationships. Given a set of nodes & connections, which can abstract anything from city layouts to computer data, graph theory provides a helpful tool to quantify & simplify the many moving parts of dynamic systems. Studying graphs through a framework provides answers to many arrangement, networking, optimization, matching and operational problems. Graphs can be used to model many types of relations and processes in physical, biological, social and information systems, and has a wide range of useful applications. Graph. A graph G is a set of vertices (nodes) v connected by edges (links) e. Thus G = (v, e). Vertex (Node). A node v is a terminal point or an intersection point of a graph. It is the abstraction of a location such as a city, an administrative division, a road intersection or a transport terminal (stations, terminuses, harbors and airports). Edge (Link). An edge e is a link between two nodes. The link (i, j) is of initial extremity i and of terminal extremity j. A link is the abstraction of a transport infrastructure supporting movements between nodes. It has a direction that is commonly represented as an arrow. When an arrow is not used, it is assumed the link is bi- directional. Planar graph. A graph where all the intersections of two edges are a vertex. Since this graph is located within a plane, its topology is two- dimensional. This is typically the case for power grids, road and railway networks, although great care must be inferred to the definition of nodes (terminals, warehouses, cities). 2.2. Connectivity Connectivity is one of the central concepts of graph theory, from both a theoretical and a practical point of view. A graph may be related to either connected or disconnected in terms of topological space. If there exists a path from one point in a graph to another point in the same graph, then it is called a connected graph. The study of connectivity parameters of graphs is of great interest in the design of reliable and fault-tolerant interconnection or communication networks. A graph is a connected graph if, for each pair of vertices, there exists at least one single path which joins them. A connected graph may demand a minimum number of edges or vertices which are required to be removed to separate the other vertices from one another. The graph connectivity is the measure of the robustness of the graph as a network. In a connected graph, if any of the vertices are removed, the graph gets disconnected. Then the graph is called a vertex-connected graph. On the other hand, when an edge is removed, the graph becomes disconnected. It is known as an edge-connected graph. 3 Electric Circuit Diagram This section is about constructing the optimal graph of the electric circuit diagram. It also consists of the minimum and maximum estimated length of wires to be used. 3.1 Layout of Diminsil Family House Figure 1. Layout of the 1st floor of Diminsil’s House Figure 2. Layout of the 2nd floor of Diminsil’s House 3.2 Transforming electrical circuit diagram into a graph 3.2.1 Position of the vertices Figure 3. 1st floor lights and switches Figure 4. 1st floor outlets Figure 5. 2nd floor lights and switches Figure 6. 2nd floor outlets 3.2.2 Graphs of the electrical circuits The length (in cm) of each edge are estimated and it is measured using a ruler. These are the symbols for every vertices: - Circuit Breaker - Switches - Lights Figure 7. Parallel-series graph for switch and lights in the 1st floor V = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] E = [(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (1,7), (2,8), (3,9), (3,10), (4,11), (5,12), (6,13)] Note. Circuit breaker is the vertex 0 Table 1. Estimated length of edges (wires) for switch and lights in the 1st floor Edge Label Length (cm) 0,1 A 24 0,2 B 86 0,3 C 47 0,4 D 117 0,5 E 171 0,6 F 174 1,7 a 15 2,8 b 18 3,9 c 19 3,10 d 25 4,11 e 32 5,12 f 44 6,13 g 33 Total length (cm) 805 Figure 8. Parallel-series graph for switch and lights in the 2nd floor V = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] E = [(0,1), (0,2), (0,3), (0,4), (0,5), (0,6), (0,7), (1,8), (2,9), (3,10), (3,11), (4,12), (5,13), (6,14), (7,15)] Note. Circuit breaker is the vertex 0 Table 2. Estimated length of edges (wires) for switch and lights in the 2nd floor Edge Label Length (cm) 0,1 A 35 0,2 B 65 0,3 C 74 0,4 D 103 0,5 E 134 0,6 F 157 0,7 G 227 1,8 a 17 2,9 b 33 3,10 c 20 3,11 d 76 4,12 e 41 5,13 f 43 6,14 g 34 7,15 h 29 Total length (cm) 1088 Figure 9. Parallel-series graph for outlets in the 1st floor V = [0, 1, 2, 3, 4, 5, 6] E = [(0,1), (0,2), (0,3), (0,4), (0,5), (0,6)] Note. Circuit breaker is the vertex 0 Table 3. Estimated length of edges (wires) for outlets in the 1st floor Edge Label Length (cm) 0,1 a 35 0,2 b 118 0,3 c 160 0,4 e 204 0,5 f 195 0,6 g 253 Total length (cm) 965 Figure 10. Parallel-series graph for outlets in the 2nd floor V = [0, 1, 2, 3, 4] E = [(0,1), (0,2), (0,3), (0,4)] Note. Circuit breaker is the vertex 0 Table 4. Estimated length of edges (wires) for outlets in the 2nd floor Edge Label Length (cm) 0,1 a 87 0,2 b 138 0,3 c 144 0,4 e 184 Total length (cm) 553 Figure 11. Series graph for switch and lights in the 1st floor V = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] E = [(0,1), (1,2), (1,3), (1,7), (2,4), (2,8), (3,9), (3,10), (4,5), (4,11), (5,6), (5,12), (6,13)] Note. Circuit breaker is the vertex 0 Table 5. Estimated length of edges (wires) for switch and lights in the 1st floor Edge Label Length (cm) 0,1 a 24 1,2 b 62 1,3 c 23 2,4 d 31 4,5 e 54 5,6 f 3 1,7 g 15 2,8 h 18 3,9 i 19 3,10 j 25 4,11 k 32 5,12 l 44 6,13 m 33 Total length (cm) 383 Figure 12. Series graph for switch and lights in the 2nd floor V = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] E = [(0,1), (1,2), (1,3), (1,8), (2,4), (2,9), (3,10), (3,11), (4,5), (4,12), (5,6), (5,13), (6,7), (6,14), (7,15)] Note. Circuit breaker is the vertex 0 Table 6. Estimated length of edges (wires) for switch and lights in the 2nd floor Edge Label Length (cm) 0,1 a 35 1,2 b 30 1,3 c 39 2,4 d 38 4,5 e 31 5,6 f 23 6,7 g 70 1,8 h 17 2,9 i 33 3,10 j 20 3,11 k 76 4,12 l 41 5,13 m 43 6,14 n 34 7,15 o 29 Total length (cm) 559 Figure 13. Series graph for outlets in the 1st floor V = [0, 1, 2, 3, 4, 5, 6] E = [(0,1), (1,2), (2,4), (2,5), (3,4), (5,6)] Note. Circuit breaker is the vertex 0 Table 7. Estimated length of edges (wires) for outlets in the 1st floor Edge Label Length (cm) 0,1 a 35 1,2 b 83 2,4 c 42 4,3 e 44 4,5 f 35 5,6 g 58 Total length (cm) 297 Figure 14. Series graph for outlets in the 2nd floor V = [0, 1, 2, 3, 4] E = [(0,4), (1,2), (2,3), (3,4)] Note. Circuit breaker is the vertex 0 Table 8. Estimated length of edges (wires) for outlets in the 2nd floor Edge Label Length (cm) 0,1 a 87 1,2 b 51 1,4 c 57 2,3 e 43 Total length (cm) 238 3.3 Observation on the electrical circuit diagram List of observations of the graphs: 1. All presented graphs are connected because there is a path between the vertices 2. If one of the vertices in a series circuit disconnects with the other, all vertices will not receive any electricity. 3. If one of the vertices in a parallel circuit is damaged, it does not affect other switch/light vertices because they are not connected with each other and their main electricity is coming from the circuit breaker. 4. Series circuit uses less wires than the parallel circuit. 3.4 Optimal Electrical Circuit The two (2) kinds of electric circuits are very important in building a house. The optimal electrical circuit for switch and lights is the parallel circuit because we can easily determine where the damaged area is, while the optimal electric circuit for outlets is the series circuit because it uses less wires (Diminsil, 2021). IV. Summary and Conclusion Summary of Findings Since the diagrams show that in the presented graphs there is a path between the vertices that are connected in Diminsil’s house. When one of the wires becomes disconnected in the graphs in a series circuit, all of the vertices lose power as well. Although their connections are not interconnected and their main electricity is coming from the circuit breaker, if one of the vertices in a parallel circuit is destroyed, it has no effect on the other switch/light vertices. A series circuit, on the other hand, utilizes fewer wires than a parallel circuit. Conclusion The Application of Connectivity in Electric Circuits is a study that can help the intended house owner to prevent power overloading and other circuit problems. The series circuit utilizes fewer wires than a parallel circuit. In comparison to a parallel circuit, a series circuit uses fewer wires. To create the graph connections for Diminsil's residence both parallel and series circuits needed to formulate the diagram. References Abdullahi, S.. (2014). An Application of Graph Theory to the Electrical Circuit Using Matrix Method. IOSR Journal of Mathematics (IOSR-JM), 10(2), 164–166. Flovik, V. (2020). What is Graph Theory, and why should you care? Towards Data Science. from https://towardsdatascience.com/what-is-graph-theory-and-whyshould-you-care-28d6a715a5c2 Gross, J. L., Yellen, J., & Zhang, P. (2013). Handbook of Graph Theory (Discrete Mathematics and Its Applications 83) (2nd ed.). Chapman and Hall/CRC. Lozin, V. (2012). Graph Theory Notes. Rodrigue, J. (2020). The Geography of Transport Systems (5th ed.). Routledge.
