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1 | | -Minimum spanning tree |
2 | | -"A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, |
3 | | -edge-weighted (un)directed graph that connects all the vertices together, |
4 | | -without any cycles and with the minimum possible total edge weight." |
| 1 | +## Minimum spanning tree ## |
| 2 | + |
| 3 | +A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of |
| 4 | +a connected, edge-weighted (un)directed graph that connects all the vertices together, |
| 5 | +without any cycles and with the minimum possible total edge weight. |
| 6 | + |
| 7 | +That is, it is a spanning tree whose sum of edge weights is as small as possible. |
| 8 | + |
| 9 | +<p align="center"> |
| 10 | + <img src="https://upload.wikimedia.org/wikipedia/commons/d/d2/Minimum_spanning_tree.svg"/> |
| 11 | +</p> |
| 12 | + |
| 13 | +> Properties : |
| 14 | +* A connected graph G can have more than one spanning tree. |
| 15 | +* All possible spanning trees of graph G, have the same number of edges and vertices. |
| 16 | +* Removing one edge from the spanning tree will make the graph disconnected, i.e. the spanning tree is minimally connected. |
| 17 | +* Adding one edge to the spanning tree will create a circuit or loop, i.e. the spanning tree is maximally acyclic. |
| 18 | +* A spanning tree does not have cycles and it cannot be disconnected. |
| 19 | + |
| 20 | +> Mathematical Properties of Spanning Tree : |
| 21 | +* Spanning tree has n-1 edges, where n is the number of nodes (vertices). |
| 22 | +* From a complete graph, by removing maximum e - n + 1 edges, we can construct a spanning tree. |
| 23 | +* A complete undirected graph can have maximum n^(n-2) number of spanning trees. |
| 24 | + |
| 25 | +> Application of Minimum spanning tree :- |
| 26 | +* Design of networks in telephone, electrical, hydraulic, TV cable, computer, road etc. |
| 27 | +* Cluster Analysis |
| 28 | +* Traveling salesman problem |
| 29 | +* Handwriting recognition |
| 30 | + |
| 31 | +> There are two most important & famous spanning tree algorithm : |
| 32 | +1. Kruskal's Algorithm |
| 33 | +2. Prim's Algorithm |
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