Hamiltonian Cycle Existence in Sparse Graphs
Consider a small graph G with vertices V = {A, B, C, D} and edges E = {A-B, B-C, C-D, D-A, A-C}. (a) Investigate the adjacency properties of the graph and discuss how the presence of these edges impacts the possibility of forming a Hamiltonian cycle. (b) Based on your analysis in part (a), prove […]
Implementing Prim’s Algorithm on an Adjacency Matrix
Given a weighted undirected graph represented by an adjacency matrix, explain how Prim’s algorithm is used to find the Minimum Spanning Tree (MST) of the graph. In your answer, describe the steps of the algorithm and how the matrix representation is utilized to determine the next edge to include.
Social Network Adjacency Analysis
A social networking site collects data on friend connections among users. The data is represented in an adjacency matrix, where each entry \(a_{ij}\) is 1 if user \(i\) is connected with user \(j\) and 0 otherwise. Note that the matrix is symmetric and no user is friends with themselves. In a small network of 5 […]
Determining Graph Connectivity from an Adjacency Matrix
Consider the following adjacency matrix representing a graph with 5 vertices: 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 Determine whether the graph is fully connected or partially connected. Justify your answer by analyzing the paths between vertices.
Basic Weighted Adjacency Matrix
Consider a graph with 4 vertices. Construct a weighted adjacency matrix where the weight between vertices i and j is given by w(i,j) = i + j if they are connected, and 0 otherwise. Assume that the graph is complete (i.e., every pair of distinct vertices is connected). Provide the resulting matrix.
Comparing Adjacency Matrix and Table Representations
Explain two key differences between an adjacency matrix and an adjacency table in representing a graph. In your answer, discuss the implications of these differences in terms of space and computational efficiency.
Basic Adjacency Matrix Definition
Define an adjacency matrix. In your response, explain how an adjacency matrix represents a simple, unweighted graph.
Constructing an Adjacency Matrix from Real-World Data
Given a dataset that represents connections between various locations with corresponding weights, perform the following tasks: (a) Convert the dataset into a node-edge list format. The dataset is provided as follows: [(A, B, 3), (B, C, 5), (A, C, 2), (C, D, 4)]. (b) Using the node-edge list from part (a), construct a weighted adjacency […]
Graph Connectivity and Cycle Detection via Adjacency Matrix
Consider the following 4×4 adjacency matrix for an undirected graph: [[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 0, 1], [1, 0, 1, 0]] (a) Determine whether the graph is connected. Justify your answer based solely on the matrix entries. (b) Identify a cycle in the graph if one exists.
Degree Calculation from an Adjacency Matrix
Given the following adjacency matrix representing an undirected simple graph without self-loops: \(\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{pmatrix}\) Compute the degree of each vertex. Express your answer as an ordered triple (degree of vertex 1, vertex 2, vertex 3).
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