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- The girth of a graphs is the length of one of its (if any) shortest graph cycles.
mathworld.wolfram.com/Girth.html
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Oct 10, 2024 · The girth of a graphs is the length of one of its (if any) shortest graph cycles. Acyclic graphs are considered to have infinite girth (Skiena 1990, p. 191). The girth of a graph may be found using Girth [g] in the Wolfram Language package Combinatorica` .
- Girth
Graph Visualization; Computation on Graphs; Graph...
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- Introduction to Girth of A Graph
- Real-World Examples and Scenarios
- Real-World Scenario and Technical Problem
- Problem Statement and Formal Definition
- Tying The Problem Statement with The Real-World Scenario
- Solution to The Problem
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The girth of a graph is the length of the shortest cycle in the graph. In other words, it is the smallest number of edges that form a closed loop. Calculating the girth of a graph is an important task in graph theory, as it helps us understand the graph's structure and properties.
Calculating the girth of a graph has multiple applications in real-world scenarios. Some examples include: 1. In transportation networks, the girth of a graph can help identify potential traffic bottlenecks or areas with a higher likelihood of congestion. 2. In social network analysis, the girth can provide insights into the structure of social net...
Let's consider a transportation network in a city. The city's road network can be modeled as a graph, where intersections are represented as vertices and roads as edges. The city officials want to identify the areas with potential traffic bottlenecks to plan for road expansions or new infrastructure projects.
Given a graph G(V, E), where V is the set of vertices representing intersections, and E is the set of edges representing roads, find the girth of the graph, i.e., the length of the shortest cycle in the graph.
By solving this problem, we can determine the smallest number of roads that form a closed loop in the city's road network. This information can then be used by city officials to identify potential traffic bottlenecks and plan for road expansions or new infrastructure projects.
We can solve this problem using a Breadth-First Search (BFS) algorithm. The BFS algorithm explores the vertices of a graph in breadthward motion, visiting all the neighbors of a vertex before moving on to the next level of vertices. To find the girth of a graph, we can perform BFS from each vertex in the graph and keep track of the shortest cycle f...
Initialize the girth as infinity.For each vertex in the graph, perform BFS and keep track of the shortest cycle found.Update the girth if a shorter cycle is found.Let's consider a sample road network represented as an adjacency list: Calculate the girth of the road network:
The bfsfunction takes the graph and a source vertex as input and performs BFS. It keeps track of the distances from the source vertex to all other vertices. When a cycle is detected, the length of the cycle is returned. Otherwise, infinity is returned. The girthfunction calculates the girth of the graph by iterating through all vertices and perform...
The same solution can be applied to other real-world problems where the girth of a graph is relevant, such as social network analysis or computer networks. By calculating the girth of the graph, we can gain insights into the structure of the network and identify potential areas for improvement or optimization.
Size of a graph is the number of edges in the graph. Create some graphs of your own and observe its order and size. Do it a few times to get used to the terms. Now clear the graph and draw some number of vertices (say n n). Try to achieve the maximum size with these vertices.
- Length – Length of the graph is defined as the number of edges contained in the graph. Length of the graph: 8. AB, BC, CD, DE, EF, FA, AC, CE.
- The distance between two Vertices – The distance between two vertices in a graph is the number of edges in a shortest or minimal path. It gives the available minimum distance between two edges.
- Eccentricity of graph – It is defined as the maximum distance of one vertex from other vertex. The maximum distance between a vertex to all other vertices is considered as the eccentricity of the vertex.
- Diameter of graph – The diameter of graph is the maximum distance between the pair of vertices. It can also be defined as the maximal distance between the pair of vertices.
- Distance between Two Vertices. It is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.
- Eccentricity of a Vertex. The maximum distance between a vertex to all other vertices is considered as the eccentricity of vertex. Notation − e(V) The distance from a particular vertex to all other vertices in the graph is taken and among those distances, the eccentricity is the highest of distances.
- Radius of a Connected Graph. The minimum eccentricity from all the vertices is considered as the radius of the Graph G. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.
- Diameter of a Graph. The maximum eccentricity from all the vertices is considered as the diameter of the Graph G. The maximum among all the distances between a vertex to all other vertices is considered as the diameter of the Graph G.
The size of a graph is its number | E | of edges, typically denoted by m. However, in some contexts, such as for expressing the computational complexity of algorithms, the term size is used for the quantity | V | + | E | (otherwise, a non-empty graph could have size 0).
The girth of a graph is the length of its shortest cycle.
