Part b
Using the sorted edge list from part (a), apply Kruskal’s Algorithm step-by-step to determine the edges included in the MST. Identify and skip any edge that would create a cycle. List the edges of the MST.
Tree Identification and Minimal Edges in Connected Graphs
Consider a connected graph where the number of vertices is \(n\) and the number of edges is \(n-1\). This property is a characteristic feature of trees. Answer the following: (a) Identify why a connected graph with 8 vertices and 7 edges must be acyclic. Describe the characteristics confirming its acyclic nature. (b) Prove that in […]
Comparing Spanning Trees and Subtrees
Consider a connected graph \(G\). Define a spanning tree and a subtree of \(G\) and explain the differences between them. Furthermore, discuss a practical scenario in network design where a spanning tree is preferred over a general subtree, especially in the context of optimizing routes and minimizing costs.
Identifying Tree Structures in Graph Theory
A tree is defined as an acyclic connected graph. Explain why a tree with \(n\) vertices must always have exactly \(n-1\) edges.
Matrix-Based Graph Analysis for Connectivity
Consider a directed graph representing a network of 5 cities. The connections between the cities are recorded in an adjacency matrix A where the entry \(a_{ij}\) is 1 if there is a direct route from city i to city j, and 0 otherwise. The adjacency matrix A is given as follows: \( A = \begin{bmatrix} […]
Exploring Euler’s Formula in Planar Graph Networks
A city’s transportation network is modeled as a connected planar graph. This network has 12 major intersections (vertices) and 17 roads (edges). Using Euler’s formula, determine the number of regions (faces) this network divides the city into. State Euler’s formula and explain why it is applicable to this problem.
Part a
Design a connected weighted graph to model an urban transportation network. Describe how you would apply a Minimum Spanning Tree (MST) algorithm (such as Kruskal’s or Prim’s) to ensure that all intersections are connected with the minimum total construction cost. Include details on how nodes and weighted edges are defined in your model.
Tracing Eulerian Circuits in Graphs
Consider a connected graph representing a simplified urban transit network where intersections are vertices and roads are edges. (a) Sketch a possible graph that includes at least 6 vertices and contains exactly two vertices of odd degree. (b) Determine whether this graph has an Eulerian path, providing justification based on Eulerian criteria.
Criteria for Eulerian Paths and Circuits
State the necessary and sufficient conditions for the existence of an Eulerian path and an Eulerian circuit in a finite, undirected graph.
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