Module Eight Homework: General Before Beginning ✓ Solved

Module Eight Homeworkgeneral Before Beginning This Homework Be Sure

Module Eight Homeworkgeneral Before Beginning This Homework Be Sure

Before beginning this homework, students should thoroughly review the corresponding textbook readings and module notes to ensure a clear understanding of the concepts involved. Additional practice is highly recommended through ungraded examples and sample problems related to the assignment topics, which are directly linked to the graded homework problems. Students are encouraged to work through those examples and post any questions to the weekly discussion forum dedicated to ungraded questions. The assignment should be completed within this document, with all steps clearly shown to demonstrate the reasoning process used to arrive at solutions.

Sample Paper For Above instruction

In this assignment, students are tasked with analyzing and drawing graphs based on specific criteria, determining properties of graphs such as the existence of Euler circuits and Euler paths, and applying Fleury’s algorithm to find an Euler circuit. These problems focus on concepts from graph theory, including graph representation, degrees of vertices, and traversal algorithms.

Problem 1: Drawing a Graph G = (V, E)

Given the vertex set V = {t, u, v, w, z} and edge set E = {e1, e2, e3, e4, e5}, with the specific edges:

  • (e1) = {v, w}
  • (e2) = {t, u}
  • (e3) = {u, v}
  • (e4) = {v, w}
  • (e5) = {t, v}

To construct this graph, first plot the vertices t, u, v, w, and z on the plane. Connect the vertices with edges according to the given sets:

  • Connect v and w with an edge (e1)
  • Connect t and u with an edge (e2)
  • Connect u and v with an edge (e3)
  • Connect v and w with another edge (e4), indicating multiple edges between v and w if the graph permits multigraphs or one edge if a simple graph is assumed
  • Connect t and v with an edge (e5)

This visual representation helps understand the structure and connectivity of graph G, essential for analyzing properties such as degrees of vertices and pathways.

Problem 2: Euler Circuit and Euler Path Analysis

Determine if the provided graphs have an Euler circuit, an Euler path, both, or neither. Recall:

  • An Euler circuit is a closed trail that uses every edge exactly once and starts and ends at the same vertex.
  • An Euler path is a trail (not necessarily closed) that uses each edge exactly once.

To analyze a graph, examine the degree (number of edges incident to each vertex). For a graph to have an Euler circuit, all vertices must have even degrees. For a graph to have an Euler path (but not a circuit), exactly two vertices can have odd degrees.

Applying this criterion to the graph in problem 1, suppose the degrees are as follows:

  • deg(t) = 1 (connected to v)
  • deg(u) = 2 (connected to t and v)
  • deg(v) = 4 (connected to w, u, w, t)
  • deg(w) = 2 (connected to v and v)
  • deg(z) = 0 (no edges)

Since t has an odd degree (1), the graph does not have an Euler circuit. Since only one vertex has an odd degree, it cannot have an Euler path either, which requires exactly two vertices of odd degree. Therefore, the graph has neither an Euler circuit nor an Euler path.

Problem 3: Using Fleury’s Algorithm to Find an Euler Circuit

Given a connected graph with an Euler circuit, Fleury’s algorithm provides a method to find such a circuit by systematically choosing edges that are not bridging edges unless no alternative exists. Starting at vertex A:

  1. Identify all edges incident to A.
  2. Select an edge that is not a bridge if possible.
  3. Traverse that edge, remove it from the graph, and move to the connected vertex.
  4. Repeat the process, always selecting non-bridge edges when possible, until all edges are traversed, ending back at the starting vertex.

Applying Fleury’s algorithm involves careful analysis of the graph's structure, often visualized or supported by a step-by-step diagram. For example, suppose the graph includes vertices connected in a way that allows an Euler circuit, and starting at vertex A, edges are labeled in the order of traversal: A–B, B–C, C–A, etc., until all edges are used.

This process yields a sequence of edges forming an Euler circuit, which is crucial for solving problems in routing, network design, and circuit analysis.

Conclusion

This assignment deepens understanding of fundamental concepts in graph theory, including graph construction, degrees of vertices, and Eulerian properties. Applying algorithms such as Fleury’s enhances problem-solving skills and practical understanding of traversal methods in networks and graphs.

References

  • Biggs, N. (1993). Discrete Mathematics. Oxford University Press.
  • Diestel, R. (2017). Graph Theory (5th ed.). Springer.
  • Harary, F. (1969). Graph Theory. Addison-Wesley.
  • West, D. B. (2001). Introduction to Graph Theory. Prentice Hall.
  • Gross, J. L., & Yellen, J. (2005). Graph Theory and Its Applications. CRC Press.
  • Bondy, J. A., & Murty, U. S. R. (2008). Graph Theory. Springer.
  • Clark, W. E. (2018). Graph Theory and Combinatorics. CRC Press.
  • West, D. B. (2001). Introduction to Graph Theory. Prentice Hall.
  • Harary, F. (1969). Graph Theory. Addison-Wesley.
  • Diestel, R. (2017). Graph Theory (5th ed.). Springer.

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