To Deliver Mail In A Particular Neighborhood The Postal Carr

To Deliver Mail In A Particular Neighborhood The Postal Carrier Ne

To deliver mail in a particular neighborhood, the postal carrier needs to walk along each of the streets with houses (the dots). Create a graph with edges showing where the carrier must walk to deliver the mail. Suppose that a town has 7 bridges as pictured below. Create a graph that could be used to determine if there is a path that crosses all bridges once. The table below shows approximate driving times (in minutes, without traffic) between five cities in the Dallas area. Create a weighted graph representing this data. Shown in the table below are the one-way airfares between 5 cities. Create a graph showing this data. Find the degree of each vertex in the graph below. Which of these graphs are connected? Travel times by rail for a segment of the Eurail system are shown below with travel times in hours and minutes. Find the path with the shortest travel time from Bern to Berlin by applying Dijkstra’s algorithm.

Paper For Above instruction

Introduction

Graphs are fundamental tools in discrete mathematics and computer science, used extensively for modeling various real-world problems such as mail delivery routes, infrastructure planning, transportation networks, and connectivity analysis. They provide a visual and mathematical representation that helps optimize routes, determine network robustness, and solve shortest path problems. This paper addresses each of the described scenarios, illustrating how to create relevant graphs, analyze their properties, and apply algorithms like Dijkstra’s for shortest path problems, supported by real-world examples and appropriate graph theory concepts.

Graph for Postal Mail Delivery in a Neighborhood

The first scenario involves creating a graph to model the streets in a neighborhood for mail delivery. Houses are represented as vertices, while the streets between houses are the edges. The main goal is to optimize the walking route so that the postal carrier can deliver mail efficiently, possibly minimizing the total distance walked or ensuring every house is reached with a minimal number of crossings.

Constructing this graph involves mapping the neighborhood's layout. Vertices correspond to houses, and edges connect houses directly accessible by walking paths. For example, if House A is connected to Houses B and C, then the graph has edges A-B and A-C. If some streets intersect or form loops, these are represented by cycles in the graph. To optimize the delivery route, algorithms like the Traveling Salesman Problem (TSP) heuristics or Minimum Spanning Trees (MST) can be employed to find efficient paths.

Graph for Bridge Crossing

The second scenario involves a town with seven bridges, which is a classic problem in graph theory dating back to the famous Seven Bridges of Königsberg. Here, each landmass or island is modeled as a vertex, and each bridge as an edge linking the vertices. The goal is to determine if there's a path crossing each bridge exactly once, which equates to finding an Eulerian Path or Circuit.

Constructing the graph involves identifying vertices for each landmass and drawing edges for each bridge. According to Euler’s theorem, such a path exists if and only if the graph is connected and exactly zero or two vertices have an odd degree. The graph can thus be used to verify whether such a path exists, informing the design of an efficient crossing route.

Weighted Graph for Dallas Area Driving Times

The third scenario uses the data of approximate driving times among five cities in the Dallas area. Vertices represent cities, and weighted edges represent the driving time in minutes between them. This model enables analyzing the shortest or fastest routes, critical for logistics, travel planning, and traffic management.

The weighted graph construction involves creating vertices for each city and drawing edges with weights corresponding to the travel times. For example, if City A and City B are 20 minutes apart, the edge A-B has weight 20. Symmetric edges indicate bidirectional travel times. Algorithms like Dijkstra’s can then be applied to find the most time-efficient routes between any two cities.

Graph for One-Way Airfares between Cities

In the case of one-way airfares between five cities, the graph is directed, with vertices representing cities and edges indicating flight routes. Each directed edge has an associated cost, representing the airfare.

This model aids in understanding travel costs and planning economical routes. For example, if the fare from City A to City B is $200, a directed edge from A to B with weight 200 is introduced. Optimizing travel in such a network involves finding shortest path or minimum cost routes using algorithms suited for directed graphs, such as Dijkstra’s algorithm or Bellman-Ford.

Vertex Degree and Connectivity Analysis

The degree of a vertex in an undirected graph refers to the number of edges incident to it. Calculating the degree involves counting how many direct connections each vertex has—for example, a house connected to three streets has degree three. Identifying whether a graph is connected involves checking if there is a path between every pair of vertices, which can be determined via traversal algorithms like BFS or DFS.

Analyzing the degree distribution and connectivity helps understand the robustness of networks, identify critical nodes, and improve design for efficiency or redundancy.

Shortest Path Using Dijkstra’s Algorithm in Eurail System

The final scenario involves finding the shortest travel time path from Bern to Berlin using Eurail rail times. Applying Dijkstra’s algorithm, which finds the shortest path in weighted graphs with non-negative weights, allows for optimal route planning.

The process includes initializing distances, selecting the vertex with the smallest tentative distance, and updating neighboring distances iteratively until reaching Berlin. The resulting path minimizes total travel time, supporting efficient travel planning.

Conclusion

Graph theory provides powerful tools for modeling and solving problems related to transportation, infrastructure, and connectivity. By constructing appropriate graphs—for neighborhood layouts, bridge crossings, city travel times, and airfares—analysts can apply algorithms like Dijkstra’s or verify Eulerian paths to optimize routes and analyze network properties. These methods, underpinned by fundamental principles such as vertex degree and connectivity, are essential in various practical applications, including logistics, urban planning, and transportation logistics.

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