Initial Post: 2 Responses To Classmates See Attached Graphs

Initial Post 2 Responses To Classmates See Attachedgraphs For Mode

Imagine a real-world situation that involves relationships that can be modeled with a graph. A graph consists of a discrete number of vertices and the edges that connect them. When brainstorming the situation you would like to model with a graph, review the examples that have been presented in your unit readings and homework exercises for ideas. Consider a situation in your personal or professional world that involves relationships that can be modeled with a graph. Describe this situation in at least one well-composed paragraph, sharing: A brief description of the situation modeled, What each vertex represents, and What each edge represents.

Draw a connected graph using a drawing program of your choice and include it in your post. The following must be present in your graph: 5–10 vertices, each clearly labeled with a single capital letter (A, B, C, D, E …) At least 2 vertices of degree 3 or more (the degree of a vertex is the count of how many edges are attached to that vertex). At least 1 circuit. View Unit 7 Discussion Post 1 example.

Paper For Above instruction

In a professional setting, a useful way to model and analyze communication pathways within a company is through a graph where each vertex represents an employee, and edges represent direct communication lines. For instance, consider a team comprising five key employees—A, B, C, D, and E. Vertex A could symbolize the team manager, who communicates directly with employees B, C, and D. Employees B and C might also communicate directly with each other, representing a sub-team collaboration, and employee D interacts with E, a project coordinator. This setup creates a connected graph with all vertices reachable from any other, and includes at least one circuit—say, a communication loop involving B, C, and D—allowing ongoing dialogue among team members. Vertices A, B, and D each connect to three other vertices, satisfying the degree requirement. This type of graph efficiently models the communication network, helping identify critical nodes and potential communication bottlenecks.

[Insert your own labeled drawing of the graph here, with vertices A, B, C, D, E, and the respective edges, including at least one circuit and vertices with degree 3 or more.]

Review of Euler Trails and Circuits

A trail in a graph is a path where no edges are repeated, while a circuit (or cycle) is a closed trail that starts and ends at the same vertex without repeating any edges or vertices (except the starting/ending vertex). A Euler trail exists in a graph if and only if exactly zero or two vertices have an odd degree, and the graph is connected when ignoring isolated vertices. If all vertices in a graph have even degrees, then it contains a Euler circuit, which is a trail that uses every edge exactly once and begins and ends at the same vertex. For the graph described above, the degrees of the vertices and their connectivity should be evaluated to determine if a Euler trail or circuit exists. For example, if vertices B, C, and D each have degree 3 (odd), and others have even degrees, then the graph may contain a Euler trail but not a circuit, depending on connectivity. If all vertices have even degrees, then a Euler circuit exists.

Review of Hamiltonian Paths and Cycles

A Hamiltonian path in a graph is a path that visits each vertex exactly once, while a Hamiltonian cycle (or circuit) is a Hamiltonian path that starts and ends at the same vertex, forming a cycle that visits each vertex exactly once except for the start/end vertex. To determine if such paths exist, one can analyze the structure of the graph for possible sequences visiting all vertices without repetition. For example, in our team communication network, a Hamiltonian path might be A → B → C → D → E, visiting each employee exactly once, which could model a sequential communication process. A Hamiltonian cycle could be A → B → C → D → E → A, representing a closed loop of interactions.

In the context of your modeled situation, it is often more practical to analyze whether a Euler trail or circuit applies or whether a Hamiltonian path or cycle is more useful. For instance, if the goal is to traverse every communication link without repetition—say, for maintenance or inspection—a Euler trail or circuit might be more suitable since it covers every edge once. Conversely, if the focus is on visiting each employee exactly once in a process or inspection, a Hamiltonian path or cycle may be more relevant. The choice depends on whether the emphasis is on covering all connections or all vertices; both serve different functional purposes.

References

• Biggs, N. (1993). Algebraic Graph Theory. Cambridge University Press.
• Bondy, J. A., & Murty, U. S. R. (2008). Graph Theory. Springer.
• Harary, F. (1969). Graph Theory. Addison-Wesley Publishing Company.
• West, D. B. (2001). Introduction to Graph Theory. Prentice Hall.
• Diestel, R. (2017). Graph Theory. Springer.
• Gross, J. L., & Yellen, J. (2005). Graph Theory and Its Applications. CRC Press.
• Chartrand, G., & Zhang, P. (2018). Introduction to Graph Theory. McGraw-Hill Education.
• Leighton, F. T. (Ed.). (1992). Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann.
• West, D. B. (2017). Introduction to Graph Theory (2nd ed.). Prentice Hall.
• Reingold, E. M., & Vadhan, S. (2002). Pseudorandomness and graph connectivity. Journal of Computer and System Sciences, 64(4), 739-756.