Minimum spanning tree cost of given Graphs

Given an undirected graph of V nodes (V > 2) named V₁, V₂, V₃, ..., V_n. Two nodes V_i and V_j are connected to each other if and only if 0 < | i - j | ? 2. Each edge between any vertex pair (V_i, V_j) is assigned a weight i + j. The task is to find the cost of the minimum spanning tree of such graph with V nodes.

Examples: 
 

Input: V = 4 
 

👁 Image

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)^th and (i - 2)^th node and the minimum spanning tree will only include the node with the minimum weight so the newly added edge will have the weight 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 = V² - V + 1 where V is the total nodes in the graph.

Below is the implementation of the above approach: 

21

Complexity Analysis:

• Time Complexity: O(1)
• Auxiliary Space: O(1)