# Minimum spanning tree cost of given Graphs

Given an undirected graph of **V** nodes (V > 2) named V_{1}, V_{2}, V_{3}, …, V_{n}. Two nodes **V _{i}** and

**V**are connected to each other if and only if

_{j}**0 < | i – j | ≤ 2**. Each edge between any vertex pair

**(V**is assigned a weight

_{i}, V_{j})**i + j**. The task is to find the cost of the minimum spanning tree of such graph with

**V**nodes.

**Examples:**

Input:V = 4

Output:13

Input:V = 5

Output:21

**Approach:** Starting with a graph with minimum nodes (i.e. 3 nodes), the cost of the minimum spanning tree will be 7. Now for every node **i** starting from the fourth node which can be added to this graph, **i ^{th}** node can only be connected to

**(i – 1)**and

^{th}**(i – 2)**node and the minimum spanning tree will only include the node with the minimum weight so the newly added edge will have the weight

^{th}**i + (i – 2)**.

So addition of fourth node will increase the overall weight as 7 + (4 + 2) = 13

Similarly adding fifth node, weight = 13 + (5 + 3) = 21

…

For n^{th}node,weight = weight + (n + (n – 2)).

This can be generalized as **weight = V ^{2} – V + 1** where

**V**is the total nodes in the graph.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function that returns the minimum cost ` `// of the spanning tree for the required graph ` `int` `getMinCost(` `int` `Vertices) ` `{ ` ` ` `int` `cost = 0; ` ` ` ` ` `// Calculating cost of MST ` ` ` `cost = (Vertices * Vertices) - Vertices + 1; ` ` ` ` ` `return` `cost; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `V = 5; ` ` ` `cout << getMinCost(V); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java implementation of the approach ` `class` `GfG ` `{ ` ` ` `// Function that returns the minimum cost ` `// of the spanning tree for the required graph ` `static` `int` `getMinCost(` `int` `Vertices) ` `{ ` ` ` `int` `cost = ` `0` `; ` ` ` ` ` `// Calculating cost of MST ` ` ` `cost = (Vertices * Vertices) - Vertices + ` `1` `; ` ` ` ` ` `return` `cost; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `int` `V = ` `5` `; ` ` ` `System.out.println(getMinCost(V)); ` `} ` `} ` ` ` `// This code is contributed by ` `// Prerna Saini. ` |

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## C#

`// C# implementation of the above approach ` `using` `System; ` ` ` `class` `GfG ` `{ ` ` ` ` ` `// Function that returns the minimum cost ` ` ` `// of the spanning tree for the required graph ` ` ` `static` `int` `getMinCost(` `int` `Vertices) ` ` ` `{ ` ` ` `int` `cost = 0; ` ` ` ` ` `// Calculating cost of MST ` ` ` `cost = (Vertices * Vertices) - Vertices + 1; ` ` ` ` ` `return` `cost; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `V = 5; ` ` ` `Console.WriteLine(getMinCost(V)); ` ` ` `} ` `} ` ` ` `// This code is contributed by Ryuga ` |

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## Python3

`# python3 implementation of the approach ` ` ` `# Function that returns the minimum cost ` `# of the spanning tree for the required graph ` `def` `getMinCost( Vertices): ` ` ` `cost ` `=` `0` ` ` ` ` `# Calculating cost of MST ` ` ` `cost ` `=` `(Vertices ` `*` `Vertices) ` `-` `Vertices ` `+` `1` ` ` ` ` `return` `cost ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `V ` `=` `5` ` ` `print` `(getMinCost(V)) ` ` ` |

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## PHP

`<?php ` `// PHP implementation of the approach ` `// Function that returns the minimum cost ` `// of the spanning tree for the required graph ` `function` `getMinCost(` `$Vertices` `) ` `{ ` ` ` `$cost` `= 0; ` ` ` ` ` `// Calculating cost of MST ` ` ` `$cost` `= (` `$Vertices` `* ` `$Vertices` `) - ` `$Vertices` `+ 1; ` ` ` ` ` `return` `$cost` `; ` `} ` ` ` `// Driver code ` `$V` `= 5; ` `echo` `getMinCost(` `$V` `); ` ` ` `#This Code is contributed by ajit.. ` `?> ` |

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**Output:**

21

## Recommended Posts:

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- Applications of Minimum Spanning Tree Problem
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