Additional Topics: Hamiltonian Paths and CyclesA Hamiltonian path in a directed graph is a path that visits every vertex in the graph exactly once.A Hamiltonian cycle in a directed graph is a cycle that visits every vertex in the graph exactly once(except for the vertex at the beginning and end of the cycle). For example, in the following directedgraph, 1,3,4,2 is a Hamiltonian path and 1, 3, 4, 2, 1 is a Hamiltonian cycle.ZFor more information about Hamiltonian paths and cycles, see zyBook section 13.7.7. (10 pt.) Given the following directed graphs:G₁Answer the following questions:• Is there a Hamiltonian path in the graph? If so, find one.• Is there a Hamiltonian cycle in the graph? If so, find one.G3

G1Hamiltonian path: Yes, the path is 1, 5, 9, 10, 6, 7, 11, 12, 8, 4, 3, 2Hamiltonian cycle: Yes,…

Answered: Additional Topics: Hamiltonian Paths…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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8 Consider the following graph: b d T/F? The directed graph has an Euler circuit. [(Click to select) The directed graph has an Euler path. [(Click to select) The Euler path is (Click to select) (Click to select) a, d, b, d, e, b, e, c, b, d ΝΑ a, d, b, d, e, b, e, c, a, b a, d, b, d, e, b, e, c, b, a a, b, d, c, e, b, e, c, b, a T/F? (Choose "NA" if there is no Euler path.)
For the directed graphs, draw the graph and show: A, A2 , A3, A4, A* , where A is the adjacency matrix for the edge set as a relation. (a) G1 = [{1, 2, 3, 4}]; (b) G2 = [{1, 2, 3, 4}]; Hint: Long matrix multiplications are not required, think about what for each of the matrices it means for a cell to contain a value of 1.
Hamiltonian clets / bath in Bipartitegraphe Show that the following graph G is bipartite by partitioning the vertices into red (R)and blue (B)sets. No need to re draw e 9 Red set: Blue set: i) Does G have a Hamiltonian circuit? ii) Does G have a Hamiltonian path? ii) For parts (i) and (ii) show a circuit or a path if it exists. If there is none, justify why not.
Consider the graph below with vertex set {a,b,c,d,e,f,g}. a Give the adjacency matrix of the graph with lexiographically ordered rows and columns.
2) Parts (a) and (b) on this page are separate. (a) An undirected simple graph has 1000 vertices of degree 6 and 200 vertices of de- gree 2. This accounts for all vertices. How many edges does it have? Work this out to a final numeric answer. (b) How many directed simple graphs (no loops or multiple edges) are there on vertex set {0, 1,...,9}? Express your answer to (b) using an appropriate formula with specific numbers plugged in, such as V100! + 2, rather than evaluating it to a final numerical result.
s. Use the Greedy and Edge-Picking algorithms to find the Hamiltonian cycle with the least total weight in the complete graph shown below. 33 57 25 72 29 65 75 89 21 50 26 45 34 60 55
Draw the edges needed in order to make the following graph complete. D E F Clear All Draw: Line Segment A Find any Hamiltonian circuit on your complete graph. Give your answer as a list of vertices, starting and ending at the same vertex. Example: ABCA
8. Knight's Tour. Here's a clever way to see how hard it is to find a Hamiltonian circuit in a graph. On a regular 8x8 chessboard, find a tour of a knight that visits each square exactly once and comes back to its starting position. Recall that a knight moves in a 2x1 L shape. Can a knight make a Hamiltonian tour on a 5x5 chessboard? How about a 4x4 chessboard? Make a conjecture as to the size of square chessboards on which a knight can completely tour.
A wheel graph W₁, n ≥ 4 is the graph made from the cycle Cn-1 by adding an additional vertex vn (in the center of the cycle) and additional edges connecting Vn to each vertex of Cn-1. Describe the graph Wn and prove that Wn is Hamiltonian.
1. (a) Find the adjacency matrix of the following graph: (c) Draw 5 5 5 5 5S V3. VA V3 V2 U1 (b) Find the incidence matrix of the graph with vertex set {1, 22, 23, 24} and edge set {1x2, 14, 223, 234, 24}. the graph whose incidence matrix is: V6 V5 1 1 1 0 00000 V2 10 0110000 0001 0 1 100 0000 1 1 010 0 1 0 0 0 V6 0 0 1 0 0 0 0 1 1 0 1 0 1
3) Let (V, E) be the graph with vertices a, b, c, d, e, f, and g, and edges ab, ac, bc, bd, be, cd, ce, de, af, df, ag, and eg. (a) Draw this graph. (b) Write down this graph's incidence table and its incidence matrix. (c) Write down this graph's adjacency table and its adjacency matrix. (d) Is this graph complete? Justify your answer. (e) Is this graph bipartite? Justify your answer. (f) Is this graph regular? Justify your answer. (g) Does this graph have any regular subgraph? Justify your answer. (h) Give an example of an isomorphism from the graph (V, E) to itself satisfying that p(a) = a. (i) Is the isomorphism from part (h) unique or can you find another isomorphism that is distinct from but also satisfies that (a) a? Justify your answer.
Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.

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