# Graph theory

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Circuit analysis is the study of methods of solving generally linear systems for unknown variables such as the voltage at a certain node or the current through a certain branch of a network. A path is a trail in which all vertices (except possibly the first and last) are distinct. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges. The degree of a vertex in a graph is the number of edges incident to it. The formula implies that in any graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. Alternatively, it is possible to use mathematical induction to prove that the number of odd-degree vertices is even, by removing one edge at a time from a given graph and using a case analysis on the degrees of its endpoints to determine the effect of this removal on the parity of the number of odd-degree…

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Answers: 1. Segment A (Position-Time on Ramp): The graph is a curve. On any graph that curves, the slope or steepness of the graph changes from one point on the graph to another. Since the slope is constantly changing, the velocity is non-constant. Also, as the graph lies above the x-axis and its slope is increasing, the velocity of the object is also increasing (speeding up) in a positive left direction. Segment B (Position-Time on Flat Surface): The graph is a straight line, which represents…

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The second source is a book which introduced the basic concepts and applications of this methodology. The authors concentrated on the causal discovery of networks and the techniques of using Bayesian networks. Furthermore, they discussed the utilisation of this methodology for causal modelling. This source is important at the initial stage of the project as it contributes to the decision-making process at the modelling technique selection stage. However, it is not as important as the other two…

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Beam search is an informed algorithm for searching a graph that extends the Breadth-First search (Jungwirth, 2006). The Breadth-First search is essentially a special-case Best-First search where the evaluation function, f(n) is the same as the heuristics function, h(n) (JONES, 2008). Therefore in order to properly discuss Beam search, there is the need for a brief introduction to Best-First algorithm. The Best-First search traverses a tree from top to bottom and searches for nodes on the same…

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Strength All the data obtained from this experiment is plotted on the graph. The accuracy and precision can be determined from the graph. Based on the graph , as the line do not pass the origin , it can be deducted that the result is fairly accurate. Since there are few outliers on the graph , the precision of this graph is moderate. Accuracy of the graph is affected by systematic errors while the precision is affected by random errors. A few measures had been taken prior in getting more…

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follow the procedure. After assembling the rover, place five weights in the ballast chamber. The amount of washers placed in the ballast is the independent variable, or what the scientist changes in an experiment. Then, turn the rover six times towards yourself, and keep count by using the sticky dot. Always turn the rover six times because this is a constant and should not be changed. Next, place the rover on the floor and let it roll. Make sure to use the same surface every time as a constant…

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INTRODUCTION Traveling Salesman Problem (TSP) is a classical combinatorial optimization problem, easy to describe but hard to solve. TSP consists of salesman and N cities, the goal of this problem is to visit each city by the salesman once with minimum tour length. The challenge in TSP is to find the route that has minimum distance. TSP can be presented via complete graph as follows: G = (V, E). Where V is the set of vertices which is equal to the number of cities and E is the set of edges…

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In each scenario mentioned in previous section, we formulate network topologies as undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with $n$ nodes and $m$ edges having demand at a node $i$ in $\mathcal{V}$ given by $d_i$. Here, $\mathcal{V}$ indicates set of nodes and $\mathcal{E}$ indicates set of edges in the topology. A link would exist in the topology between node $i$ and node $j$, if they are in the communication range of each-other satisfying constraint, \begin{equation} P_i…

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applications of graph theory in transport planning. A graph is a symbolic representation of a network and of its connectivity that has a set of nodes and a set of edges where each edge connects to two nodes. Graph representations offer a convenient means of handling the topological and associated information describing a road network and the use of graph theory in supporting network analysis and generalization. Therefore, an appropriate classification of road data must take into account…

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As I was reading the final chapter of the 5 Practices book, I was reminded of how important collaboration is – not only for my college studies but more importantly for my future teaching career. Reflecting on the work Sam & I put into our unit plan for this course, it was helpful to talk through ideas with one another and to see the different perspectives we each had. There are many times where I get stuck in my own thinking and it is truly helpful to be able to gain a fresh outlook when working…

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