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# Count of Nodes with Maximum Connection in an Undirected Graph

In the field of network analysis, the number of nodes with the highest degree, signifying the greatest number of connections to other nodes in the network, is referred to as the "count of nodes with maximum connection" in an undirected graph. The number of edges that incident upon a node determines its degree. We may determine the critical or central points in the graph by identifying the nodes with the highest degree. This has important ramifications for a variety of applications, including network research, social network studies, and optimisation methods. Understanding these crucial nodes makes it easier to comprehend the network's structural characteristics and makes it simpler to develop effective algorithms for a variety of challenging jobs, advancing network analysis and its related applications.

## Methods Used

Naive Approach

Optimization using Hash Map

## Naive Approach

In the simplistic method for "Count of nodes with maximum connection in an undirected graph," we repeatedly go through every node in the network. We count the number of edges that connect each node to determine its degree. We record the highest degree encountered during this operation and count the nodes that have that degree. The temporal complexity of this technique, despite being straightforward, is O(V + E), where V is the number of vertices (nodes) and E is the number of edges. Although efficient for small graphs, its exhaustive nature may make it ineffective for bigger graphs.

### Algorithm

Set up variables:

count maxDegree = 00 MaxNodes

traverse the graph's whole network of nodes.

Each node should have its degree initialised to 0.

Go through each node's neighbours one by one.

Percentage increase for each neighbour you visit.

Update countMaxNodes and maxDegree:

If the current node's degree is higher than maxDegree:

The current node's degree should be entered as maxDegree.

countMaxNodes should be set to 1.

If the current node's degree is equal to maxDegree:

Add one to countMaxNodes.

The outcome is countMaxNodes.

### Example

#include <iostream> #include <vector> using namespace std; int countNodesWithMaxConnection(vector<vector<int>>& graph) { int maxDegree = 0; int countMaxNodes = 0; for (int node = 0; node < graph.size(); ++node) { int degree = graph[node].size(); if (degree > maxDegree) { maxDegree = degree; countMaxNodes = 1; } else if (degree == maxDegree) { countMaxNodes++; } } return countMaxNodes; } int main() { vector<vector<int>> graph = { {1, 2, 3}, // Node 0 is connected to nodes 1, 2, and 3 {0, 2}, // Node 1 is connected to nodes 0 and 2 {0, 1}, // Node 2 is connected to nodes 0 and 1 {0}, // Node 3 is connected to node 0 }; int result = countNodesWithMaxConnection(graph); cout << "Count of nodes with maximum connection: " << result << endl; return 0; }

### Output

Count of nodes with maximum connection: 1

## Optimization using Hash Map

The "Count of nodes with maximum connection in an undirected graph" optimisation problem uses a hash map to efficiently store the degree of each node while traversing all the edges and nodes. The degree of each node is tracked in the hash map, which is updated when edges are processed. We simultaneously determine the highest degree and the number of nodes with that degree.This method makes it easier to identify the nodes in the graph with the most connections by obtaining a reduced time complexity of O(V + E), where V stands for the number of nodes and E for the number of edges.

### Algorithm

Create a blank hash map in the beginning to hold each node's degree.

traverse the graph's whole set of edges.

Update the degree in the hash map for both endpoints (nodes) for each edge.

Keep track of the highest degree discovered thus far while refreshing the hash map.

Once every edge has been processed, traverse the hash map to count the nodes with the highest degree.

As a consequence, give the total number of nodes with the greatest degree.

### Example

#include <iostream> #include <vector> #include <unordered_map> using namespace std; int countNodesWithMaxConnection(const vector<vector<int>>& graph) { unordered_map<int, int> degreeMap; // Stores node degree int maxDegree = 0; // Tracks maximum degree found for (const vector<int>& edge : graph) { for (int node : edge) { degreeMap[node]++; maxDegree = max(maxDegree, degreeMap[node]); } } int countMaxDegreeNodes = 0; // Count of nodes with maximum degree for (const auto& entry : degreeMap) { if (entry.second == maxDegree) { countMaxDegreeNodes++; } } return countMaxDegreeNodes; } int main() { vector<vector<int>> graph = { {1, 2}, {2, 3, 4}, {1, 3, 5}, {2, 4}, {1, 3, 5}, {2, 4} }; int result = countNodesWithMaxConnection(graph); cout << "Count of nodes with maximum connection: " << result << endl; return 0; }

### Output

Count of nodes with maximum connection: 1

## Conclusion

A key statistic of relevance in network research and optimisation is the "Count of nodes with maximum connection in an undirected graph." To solve this problem, we investigated the Naive Approach and Hash Map Optimisation. The Naive Approach is simple, but because of its O(V + E) time complexity, it is less suited for expansive graphs. However, although having a similar temporal complexity, hash map optimisation considerably boosts efficiency. We gain significant understanding of the intricate structural details of the graph and find the central nodes with the most connections through this procedure. Such discoveries have enormous implications for a variety of fields, including network research, social network analysis, and other optimisation methods. As a result, the search of this crucial statistic has important ramifications for understanding intricately linked systems and advancing science.