Lecture

Minimal Spanning Tree

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Description

This lecture introduces the concept of a weighted graph where each edge is assigned a numerical weight, defining the weight of a graph as the sum of weights of all its edges. The problem of finding a minimum weight spanning tree for a given weighted connected graph is addressed using the Greedy algorithm (or Kruskal's algorithm). The correctness of Kruskal's algorithm is discussed, showing that the resulting graph contains all vertices of the original graph and has no cycles. The process of adding edges to the spanning tree to ensure minimal weight is explained, along with the comparison of different spanning trees based on their weights. The lecture concludes with a detailed explanation of the algorithm's iterative procedure to obtain the minimal spanning tree.

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (25)

Kruskal's algorithm

Kruskal's algorithm (also known as Kruskal's method) finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.

Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.

Graph (discrete mathematics)

In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.

Glossary of graph theory

This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

Directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. In formal terms, a directed graph is an ordered pair where V is a set whose elements are called vertices, nodes, or points; A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.

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