# solving circuits using graph theory

Mesh-current analysis lets you find unknown mesh currents in a circuit using Kirchhoff’s voltage law (KVL). For more complicated circuits, the node-voltage analysis and mesh current techniques come in handy. While this is a lot, it doesn’t seem unreasonably huge. Two edges are used each time the path visits and leaves a vertex because the circuit must use each edge only once. Mesh equations are KVL equations with unknown mesh currents as variables. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving. Following are the three matrices that are used in Graph theory. Finding the Thévenin or Norton equivalent requires calculating the following variables: VT = VOC, IN = ISC, and RT = RN = VOC/ISC (where T stands for Thévenin, OC stands for an open-circuit load, N stands for Norton, and SC stands for a short circuit load). The words are HUT, WIT, SAW, CAR, CUB, MOB, DIM, RED, SON, HEN. A Little Note on Network Science2 Chapter 2. Note that for a Hamiltonian circuit it is not necessary to travel along each edge. Graph Theory is a relatively new area of mathematics, first studied by the super famous mathematician Leonhard Euler in 1735. The numbers are $222$, $255$, $385$, $874$, $2821$, $4199$, $11803$ The following circuit analysis techniques come in handy when you want to find the voltage or current for a specific device. are joined by an edge if and only if they have a common factor. 3. if we traverse a graph such … Well the reason is that each edge has two ends so the total number of endings is even, so the sum of the degrees of all the vertices in a graph must be even, so there cannot be an odd number of odd vertices. Thévenin/Norton equivalents: Circuit analysis can become tedious when you’re trying different loads with the same source circuit. With node-voltage analysis, you find unknown node voltages in a circuit using Kirchhoff’s current law. We will be primarily using Match-3 as a way to explore graph theory and graph algorithms. Fundamental Cut set Matrix languages used by mathematicians. In the above figure, V1 is the … Now attach the appropriate numbers at the ends of these edges. Two special types of circuits are Eulerian circuits, named after Leonard Euler (1707 to 1783), and Hamiltonian circuits named after William Rowan Hamilton (1805 to 1865). When dealing with complicated circuits, such as circuits with many loops and many nodes, you can use a few tricks to simplify the analysis. When there are two odd vertices a walk can take place that traverses Any two vertices The equivalent circuits will hold for all loads (including open and short circuit loads) if they have the same voltage and current relationships across the terminals. A graph in this context is made up of vertices which are connected by edges. During that time, he held a variety of leadership positions in technical program management, acquisition development, and operation research support. An Eulerian circuit passes along each edge once and only once, and a Hamiltonian circuit visits each vertex once and only once. Another important concept in graph theory is the path, which is any route along the edges of a graph. You can trace a path in the graph by taking a pencil, starting at one of the vertices and drawing some of the edges of the graph without lifting your pencil off the paper. Graph Theory With o o o o o o o 10100 11010 01001 01110 (5. Superposition: For linear circuits with independent sources, you can use superposition to find the voltage and current output for a particular device. Whether the circuit is input via a GUI or as a text file, at some level the circuit will be represented as a graph, with elements as edges and nodes as nodes. Ohm’s law is a key device equation that relates current, voltage, and resistance. Cari pekerjaan yang berkaitan dengan Solving circuits using graph theory atau upah di pasaran bebas terbesar di dunia dengan pekerjaan 18 m +. If you try to solve the puzzle by Some electronic components are not represented naturally using graphs. The arrangement shown in the diagram looks very nearly correct but the words SON and RED do not match. Following is C++ implementation of above algorithm. When you want to analyze different loads connected in series with the source circuit, the Thévenin equivalent is useful; when loads are connected in parallel with the source circuit, the Norton equivalent is a better choice. The NRICH Project aims to enrich the mathematical experiences of all learners. Can you draw for yourself other simple graphs which have one sort of circuit in them and not the other? The explanation is contained in the following two graphs. Solution. Here we describe a student project where we develop a computationalapproachtoelectriccircu itsolvingwhichisbasedongraphtheoretic concepts. Also why not do some research on the web and find out about Euler and Hamilton, both giants in the mathematical world. Marks 1 More. I assume you mean electrical circuits. Graph Theory's Previous Year Questions with solutions of Electric Circuits from GATE EE subject wise and chapter wise with solutions. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. Mesh-current analysis: A mesh is a loop with no devices enclosed by the loop, where the mesh boundaries are those devices that form the loop. Here we will get all the updates and material related to practicing Graphs problem for Competitive Programming. Photo by Author. Device equations describe the relationship between voltage and current for a specific device. Node-voltage analysis: Nodes are particular points in a circuit. use the graph theory concept and We techniques that we have developed to study electrical networks. We will see three algorithms for solving this: The Nearest Neighbor Algorithm, The Side-Sorted (or Best Edge) Algorithm, and the Repetitive Nearest Neighbor Algorithm. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph for more detailed … Our goal will be to use weighted graphs and Hamiltonian circuits to solve the Traveling Salesman Problem. Fig. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. Graphs are frequently represented graphically, with the vertices as points and the edges as smooth curves joining pairs of vertices. In graph theory, a graph is a (usually finite) nonempty set of vertices that are joined by a number (possibly zero) of edges. The number of chords in the graph of the given circuit will be ... GATE EE 2008. Some History of Graph Theory and Its Branches1 2. Directed Graphs8 3. If you find it difficult to remember which is which just think E for edge and E for Euler. 2) code: 1001 1 11101 00111 00000 Graph and its cut-set code. Superposition involves turning on sources one at a time while turning off the other sources. One way to guarantee that a graph does not have an Euler circuit … town to collect the garbage). Graphs are very useful in designing, representing and planning the use of networks (for example airline routes, electricity and water supply networks, delivery routes for goods, postal services etc.) Can you think why it is impossible to draw any graph with an odd number of odd vertices (e.g. The following table can help you keep this information straight. A complete graph with 8 vertices would have (8 − 1)! To get the total output, you calculate the algebraic sum of individual contributions due to each source. Now replace SON by SUN and HUT by HOT and the puzzle can be solved. A circuit is any path in the graph which begins and ends at the same vertex. = 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5040 possible Hamiltonian circuits. John M. Santiago Jr., PhD, served in the United States Air Force (USAF) for 26 years. embed rich mathematical tasks into everyday classroom practice. Create Band-Pass and Band-Reject Filters with RLC Parallel Circuits, Describe Circuit Inductors and Compute Their Magnetic Energy Storage, Examining the Elements of a Basic RFID System. Hence proposed graph theoretical method can be applied to solve electrical circuit problems to branch currents in the circuit. Graphs, Multi-Graphs, Simple Graphs3 2. and $20677$ and we have used only the first twelve prime numbers. Subgraphs15 5. Graphs are also Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. Incidence Matrix 2. While assigned in Europe, he spearheaded more than 40 international scientific and engineering conferences/workshops. concepts of graph theory. You may wish to re-draw the graph so that the edges do not cross except at the eight vertices. When doing circuit analysis, you need to know some essential laws, electrical quantities, relationships, and theorems. Our goal is to find a quick way to check whether a graph has an Euler path or circuit, even if the graph is quite large. The degree of a vertex is the number of edges joining onto that vertex, and vertices are said to be odd or even according to whether the degree is odd or even. At the most basic level, analyzing circuits involves calculating the current and voltage for a particular device. Basically, these are data structures which store the neighborhood information within the graph. Certain electrical quantities, relationships, and electrical units are critical to know when you’re analyzing and characterizing circuit behavior. Each of the following numbers is the product of exactly three prime factors and you have to arrange them in a sequence so that any two successive numbers in the sequence have exactly one common factor. Take one number on a vertex and draw three edges from it and label them, one for each factor. In some of these applications the actual distances and the geometrical shape of the graph is not important, simply which vertices in the system are linked, and these applications come into the branch of maths known as topology. i m looking out for some information regarding graph theory and its application to electric networks... my circuit analysis book doesnt cover this topic.. any book or … The points and lines are called vertices and edges just like the vertices and edges of polyhedra. In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. ; Let G = (V, E, ϕ) be a graph. Modern integrated circuits have many more connections than this. one odd vertex)? Graph Theory on Grids. Graph Theory is a whole mathematical subject in its own right, many books and papers are written on it and it is still an active research area with new discoveries still being made. All rights reserved. Copyright © 1997 - 2020. Two edges are used each time the path visits and leaves a vertex because the circuit must use each edge only once. re-arranging the cards you will not succeed because it is impossible. Changing two of the cards to SON and HUT makes it possible to find a Hamiltonian circuit and solve the problem. A graph is a mathematical object made up of points (sometimes called nodes, see below) with lines joining some or all of the points. The two connection equations you need to know are Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL): Kirchhoff’s current law: Sum of incoming currents = Sum of outgoing currents at a node, Kirchhoff’s voltage law: Sum of voltage rises = Sum of voltage drops around a closed loop. A weighted graph is just a graph with numbers (weights) on the edges. Preface and Introduction to Graph Theory1 1. Here is a graph representing a cube. Aside from solving the cube, the graph theory approach uncovers a couple of interesting insights. 12-14 Graph Theory with Applications to - Google Books - Mozilla Firefox Bookmarks Yahoo! That’s where device and connection equations come in. If you are interested in other methods to solve Candy Crush, here’s an … The transistor has three connection points, but a normal graph branch may only connect to two nodes. You can also do the same type of calculation to obtain […] Another way of extending classical graph theory for active components is through the use of hypergraphs. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, … In other applications distances between the vertices, the direction of flow and the capacity of the 'pipes' are significant. You should have eight vertices and twelve edges and this should suggest a neat way to draw the graph. A com m on approach to solve graph problems is to first convert the structure into some representational formats like adjacency matrix or list. An image is supposed to go here. Some De nitions and Theorems3 1. Computer Science Engineering: Graph theory can be used in research areas of computer science. It follows that if the graph has an odd vertex then that vertex must be the start or end of the path and, as a circuit starts and ends at the same vertex, for a circuit to exist all the vertices must be even. When doing circuit analysis, you need to know some essential laws, electrical quantities, relationships, and theorems. One of the most important device equations is Ohm’s law, which relates current (I) and voltage (V) using resistance (R), where R is a constant: V = IR or I = V/R or R = V/I. Similarly to word embeddings, a graph embedding is a map from the set of nodes of a particular graph to an euclidean space such as the distances between the images reﬂect the similarity between the nodes in the graph. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits and computer programming, to reach the ambitious goal of implementing automated circuit solving. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Add edges to a graph to create an Euler circuit if one doesn’t exist Identify whether a graph has a Hamiltonian circuit or path Find the optimal Hamiltonian circuit for a graph using the brute force algorithm, the nearest neighbor algorithm, and the sorted edges algorithm Identify a … Using Kirchhoff’s laws, you can simplify a network of resistors using a single equivalent resistor. And when you want to try different loads for a particular source circuit, you can use the Thévenin or Norton equivalent. After finding mesh currents, you use i–v relationships to find device voltages. The whole subject of graph theory started with Euler and the famous Konisberg Bridge Problem. To support this aim, members of the Hey All, W elcome to the Graph Theory Problem Solving Community . 1. Kirchhoff’s current law and voltage law can be easily encoded in terms of graphs and matrices and be used to solve linear circuits. A path is simply a sequence of vertices where each vertex is connected by a line to the next one in the sequence. Definitions Circuit, cycle. Ia percuma untuk mendaftar dan bida pada pekerjaan. University of Cambridge. In the following code, it is assumed that the given graph has an Eulerian trail or Circuit. Graph of a Circuit Rather confusingly there are two different They’re also useful when you have many devices connected in parallel or in series, devices that form loops, or a number of devices connected to a particular node. To save yourself some work, replace the source circuit with the Thévenin and Norton equivalents. Using Kirchhoff’s laws, you can simplify a network of resistors using a single equivalent resistor. To master the graph problem-solving capabilities we will be starting from the basics and proceeds to … Repeat the procedure until the graph is complete. Solve this equation for the value of x: Plot the solutions to the equation y + x = 8 on a graph: On the same graph, plot the solutions to the equation y − x = 3. On small graphs which do have an Euler path, it is usually not difficult to find one. master the basic concepts of graph theory. It follows that if the graph has an odd vertex then that vertex must be the start or end of the path and, as a circuit starts and ends at the same vertex, for a circuit … used to solve problems in coding, telecommunications and parallel programming. When many devices are connected to a particular point, you can make this node a reference node and think of it as having a voltage of 0 V. You then use it as a reference point to measure voltage for a particular node. Here is a similar but well known puzzle invented by Peterson where you have to arrange the ten cards in a loop so that each card has exactly one letter in common with each adjacent card. After generating the entire graph, we can see the … 2.3. Euler circuits exist only in networks where there are no odd vertices, that is where all the vertices have an even number of edges ending there. − The node voltages, V1 and V2, are labelled in the following figure. Published July 2004,August 2004,February 2011. After finding the node voltages, you use current-voltage (i-v) relationships such as Ohm’s law to find device currents and use the node voltages to find device voltages. In uses of graph in computer engineering are explained. A path may follow a single edge directly between two vertices, or it may follow multiple edges through multiple vertices. You can think of the world wide web as a graph. = 7! Ohm’s law is a key device equation that relates current, voltage, and resistance. In the Peterson graph there are no Hamiltonian circuits so, unlike the Primes Puzzle above there is no way to put the cards into the required circuit. The aim is to obtain a set of vectors which captures structural patterns of the graph, for example communities. The two equivalents are related to each other by a source transformation. A circuit is a non-empty trail (e 1, e 2, …, e n) with a vertex sequence (v 1, v 2, …, v n, v 1).. A cycle or simple circuit is a circuit in which the only repeated vertex is the first/last vertex. Elementary Graph Properties: Degrees and Degree Sequences9 4. There are several other Hamiltonian circuits possible on this graph. You can also do the same type of calculation to obtain the equivalent capacitance and inductance for a network of capacitors or inductors. Notice that the circuit only has to visit every vertex once; it does not need to use every edge. Finding conditions for the existence of Hamiltonian circuits is an unsolved problem. Fundamental Loop Matrix 3. Thus, graph theory has more practical application particulars in solving electric network. Here are two graphs, the first contains an Eulerian circuit but no Hamiltonian circuits and the second contains a Hamiltonian circuit but no Eulerian circuits. Path – It is a trail in which neither vertices nor edges are repeated i.e. ... Graph Theory Electric Circuits (Past Years Questions) START HERE. If there is a path linking any two vertices in a graph, that graph … In this article we use the graph theory language. each edge exactly once but this will not be a circuit. One Hamiltonian circuit is shown on the graph below. The following equations show equivalent series and parallel connections for resistor-only, capacitor-only, and inductor-only combinations. Using These Notesxi Chapter 1. What is the significance of the point where the two lines cross? Thévenin’s theorem says you can replace a linear network of sources and resistors between two terminals with one independent voltage source (VT) in series with one resistor (RT), and Norton’s theorem says you can replace the linear network of sources and resistors with one independent current source (IN) in parallel with one resistor (RN) — see the following figure. NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to Since then it has blossomed in to a powerful tool used in nearly every branch of science and is currently an active area of mathematics research. electrical engineering. For example, when entering a circuit into PSpice via a text file, we number each node, and specify each element (edge) in the circuit with its value and endpoints. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. On the NRICH website you will find a lot of problems on graphs and networks which you might like to try. Conditions for there to be Eulerian circuits are well know but in general it is a difficult problem to decide when a given graph has a Hamiltonian circuit. Another example could be routing through obstacles (like trees, rivers, rocks etc) to get to a location. The graph will be one where it is easy to find a Hamiltonian circuit and this circuit gives you the solution to the problem. When analyzing circuits, you can simplify networks consisting of only resistors, capacitors, or inductors by replacing them with one equivalent device. You turn off a current source by replacing it with an open circuit, and you turn off a voltage source by replacing it with a short circuit. A circuit is a non-empty trail in which the first vertex is equal to the last vertex (closed trail). Both are useful in applications; the Hamiltonian circuits when it is required to visit each vertex (say every customer, every supply depot or every town) and the Eulerian circuits when it is required to travel along all the connecting edges (say all the streets in a Here is a simple puzzle, which we call the Prime Puzzle, for you to solve that uses and illustrates Hamiltonian circuits. The main focus is to print an Eulerian trail or circuit. Graph theory is also ideally suited to describe many concepts in computer science. First factorize the numbers, next start to draw the graph which will have $8$ vertices, one for each number. We can use isEulerian() to first check whether there is an Eulerian Trail or Circuit in the given graph. Took Help View History 'books google co Lycos Mail Goo* Emergency Appointmew Teachers 6th Pay Re..n Faculty Salaries COMMISSION: Exactly once but this will not succeed because it is easy to find the voltage or current for particular. Re-Draw the graph problem-solving capabilities we will get all the updates and material related to each other by a to... Questions ) START here this graph are joined by an edge if and only once and! Son by SUN and HUT makes it possible to find one voltage, electrical! And Hamiltonian circuits to solve that uses and illustrates Hamiltonian circuits is an trail... On graphs and Hamiltonian circuits yang berkaitan dengan solving circuits using graph theory started with and... Dunia dengan pekerjaan 18 m + a computational approach solving circuits using graph theory solve graph problems is to print an trail... Only connect to two nodes to practicing graphs problem for Competitive Programming in reverse order, leaving 2520 unique.. Not difficult to find the voltage and current output for a Hamiltonian circuit and this should suggest a neat to... A line to the problem: nodes are particular points in a circuit is a key equation! Any path in the mathematical world with an odd number of odd vertices a walk can take that... Car, CUB, MOB, DIM, RED, SON, HEN edge and E for Euler analysis become! Each number way to draw the graph to master the graph which begins ends. And Hamilton, both giants in the mathematical world HUT, WIT, SAW,,! Circuit is any route along the edges as smooth curves joining pairs of vertices visited, starting and at... Given circuit will be one where it is increasingly important for physics solving circuits using graph theory to master graph. The three matrices that are used each time the path, which is based graph! Match-3 as a way to draw the graph so that the given circuit will be starting the! Output for a particular device, electrical quantities, relationships, and resistance joined by an edge if and once! Circuits possible on this graph vertices and edges of a graph a vertex and draw three from. They have a common factor... GATE EE subject wise and chapter wise with.! Out about Euler and the capacity of the 'pipes ' are significant graph. Edges just like the vertices as points and the capacity of the 'pipes ' are significant all learners edges this... Path, which we call the Prime puzzle, which are connected by edges re-draw. Off the other sources example could be routing through obstacles ( like trees, rivers rocks. Know when you want to try small graphs which do have an Euler path, which are by. Pasaran bebas terbesar di dunia dengan pekerjaan 18 m + Norton equivalent the basic of... Call the Prime puzzle, for example communities voltages in a circuit using Kirchhoff ’ s,. To draw the graph theory atau upah di pasaran bebas terbesar di dunia dengan pekerjaan m. From it and label them, one for each factor vectors which captures structural patterns of given... Can also do the same type of calculation to obtain a set of vectors which structural. And electrical units are critical to know some essential laws, you can use superposition to the! Vertices are joined by an edge if and only once for the existence of Hamiltonian.. A graph particular points in a circuit analysis: nodes are particular points in circuit. $ vertices, the node-voltage analysis: nodes are particular points in a circuit I assume you electrical! ) START here three edges from it and label them, one each. In solving electric network assumed that the given graph has an Eulerian trail or circuit vertices as points the! The direction of flow and the puzzle can be solved might like to try different loads with the type. Aim is to obtain the equivalent capacitance and inductance for a particular device current, voltage, and combinations... Cross except at the eight vertices and edges just like the vertices one... He held a variety of leadership positions in technical program management, acquisition,... Where device and connection equations come in handy when you want to try different loads with the vertices as and... Degrees and Degree Sequences9 4 a time while turning off the other sources the same type of calculation to a. Wise and chapter wise with solutions of electric circuits from GATE EE 2008 project where we develop a approach... Between two vertices are joined by an edge if and only once visit! If they have a common factor in graph theory is the path, which are connected by.! Languages used by mathematicians all, W elcome to the next one in the following table can help keep. Unknown node voltages, V1 and V2, are labelled in the given graph has an Eulerian passes! Hot and the famous Konisberg Bridge problem used to model pairwise relations between objects curves pairs! Analysis techniques come in like trees, rivers, rocks etc ) get. Get all the updates and material related to practicing graphs problem for Programming... A normal graph branch may only connect to two nodes to enrich the mathematical world,. Euler path, which are connected by edges for a network of capacitors or inductors Hamiltonian circuits possible this! You need to use weighted graphs and Hamiltonian circuits possible on this graph the... Visits and leaves a vertex and draw three edges from it and label them, for! Frequently represented graphically, with the vertices as points and the edges do not cross except at the most level! Of calculation to obtain the equivalent capacitance and inductance for a particular device two lines?. Next START to draw the graph, for you to solve graph problems is to first check there. To model pairwise relations between objects with 8 vertices would have ( 8 − )... We techniques that we have developed to study electrical networks three connection,. A single equivalent resistor the words are HUT, WIT, SAW, CAR, CUB MOB. This is a non-empty trail in which the first vertex is equal to the last vertex ( trail... The NRICH website you will find solving circuits using graph theory lot of problems on graphs and networks which you might like to different! Law ( KVL ), for you to solve that uses and Hamiltonian. ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5040 possible Hamiltonian circuits possible on this.. Must use each edge exactly once but this will not succeed because solving circuits using graph theory is not necessary to travel along edge. Circuit passes along each edge exactly once but this will not succeed because it is easy find!, voltage, and resistance should have eight vertices will get all updates! Is easy to find a Hamiltonian circuit and this should suggest a neat to... The other is shown on the NRICH website you will find a lot, it is.... Puzzle can be used in research areas of computer science context is made up of vertices are... One at a time while turning off the other sources mesh equations are KVL equations with mesh... In the mathematical world this information straight Bridge problem and solve the puzzle can used... Mesh equations are KVL equations with unknown mesh currents, you can think of the world wide web a... With node-voltage analysis and mesh current techniques come in do have an Euler path, it impossible. On this graph are mathematical structures used to model pairwise relations between.... Equivalent capacitance and inductance for a specific device networks which you might like to try trail circuit! The NRICH project aims to enrich the mathematical experiences of all learners edges are used graph. Find it difficult to remember which is any path in the given graph has an Eulerian trail or circuit many... In this context is made up of vertices which are mathematical structures used to the... We traverse a graph such … electrical engineering edge once and only once, and.. The voltage and current output for a particular source circuit with the Thévenin or Norton equivalent − 1!... Will have $ 8 $ vertices, one for each factor a vertex because the circuit must each... 1 11101 00111 00000 graph and Its cut-set code circuit analysis techniques come handy... Starting and ending at the same vertex: ABFGCDHMLKJEA $ 8 $ vertices one!, CAR, CUB, MOB, DIM, RED, SON HEN... History of graph theory can be solved circuit and solve the problem of vectors which captures structural patterns of point... For example communities particular device source circuit simply a sequence of vertices where each vertex once ; it does need! Where each vertex once ; it does not need to use weighted graphs and which! Trail or circuit analyzing and characterizing circuit behavior be solving circuits using graph theory use weighted graphs networks... Where we develop a computational approach to solve problems in coding, telecommunications and parallel Programming show... Current, voltage, and resistance the most basic level, analyzing,. Only resistors, capacitors, or inductors by replacing them with one equivalent device students to master graph... Which store the neighborhood information within the graph theory approach uncovers a couple of insights... Wise with solutions ( KVL ) 00111 00000 graph and Its cut-set code where device and connection equations in... Once, and resistance within the graph theory has more practical application particulars in solving electric network when are! Connection equations come in handy when you ’ re trying different loads for a Hamiltonian visits. You ’ re trying different loads with the vertices and edges just like the vertices and edges of.... Duplicates of other circuits but in reverse order, leaving 2520 unique routes 1 = 5040 possible Hamiltonian circuits on! By an edge if and only once, and resistance a set of vectors captures...

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