The provided code is an implementation of Prim's algorithm using C, which is used to find a minimum spanning tree for a weighted undirected graph I'll explain the code line by line, and afterward, we'll go through an example to demonstrate how the code works with a specific set of inputs.

First, let's go through the code:

#include <stdio.h#include <stdlib.h>

#define infinity 9999
#define MAX 20

int G][MAX], spanning[MAX][MAX], n;

1-3: Include standard input/output library for console I/O operations, and include the standard library for various functions like dynamic memory allocation. 4-5: Define constants for representing infinity (as 9, to denote a very large number) and the maximum number of vertices (MAX) that our graph have (20 in this case). 6: Declare global matrices G for the graph's adjacency matrix, spanning for storing the minimum spanning tree, and integer n for storing the number of vertices in the graph.

int prims() {

7: Begin the definition of the prims function that implements Prim's algorithm.

    int cost[MAX][MAX];
    int u, v, min_distance, distance[MAX from[MAX];
    int visited[MAX], no_of_edges,, min_cost, j;

8-9: Declare local variables and arrays: cost to store the cost of edges, u andvto store vertices,min_distanceto find the minimum edge,distanceto hold the shortest distance from the constructed spanning tree to an unvisited node,fromto track from which node we the node with minimum distance,visitedto mark nodes as visited during the algorithm's execution,no_of_edgesto count the of edges added to the spanning tree,min_costto calculate the total cost of the spanning tree and iteratorsiandj`.

    // Create cost[][] matrix, spanning[][]
    for ( = 0; i < n; i++) {
        for (j = 0; j < n j++) {
            if (G[i][j] == 0)
                cost[i][j] = infinity;
            else
                cost[i][j] = G[i][j];
            spanning][j] = 0;
        }
    }
``12-19: Initialize the `cost` matrix with the values from the adjacency matrix `G`. If there is no edge between vertices (represented by a zero in `G`), set the cost toinfinity`. Also, initialize the `spanning` matrix to zero, which will eventually represent the minimum spanning tree.

```c
    // Initialise visited[], distance[], and from[]
    distance[0] = 0;
    visited[0] = 1;
    for (i = 1; i < n; i {
        distance[i] = cost[0][i];
        from[i] =0;
        visited[i] = 0;
    }

22-28 Set the initial node's (assumed to be 0) distance to zero and mark it as ``. For the rest of the nodes, set the distance as the direct cost from node 0 and mark them as unvisited. Initialize from to track the paths.

    min_cost =0; // Cost of spanning tree
    no_of_edges = n - 1 // No. of edges to be added

30-31: Initialize min_cost to zero, representing the total cost the spanning tree. Set no_of_edges to n-1, which is the number of edges that will be included in the spanning tree for `n nodes.

    while (no_of_edges >0) {

33: Begin a while loop that will continue until all n-1 edges are added to the spanning tree.

        // Find the vertex minimum distance from the tree
        min_distance = infinity;
        for (i = 1; i < n; i++) {
            if (visited[i == 0 && distance[i] < min_distance) {
 v = i;
                min_distance = distance[i];
            }
        }
        u = from[v];

35-42 Search for the non-visited vertex with the minimum distance from set of visited vertices. Once found, the variable v will hold the index of such a vertex, andu is set to the vertex from where this minimum distance is measured (from[v]`).

        // Insert the in spanning tree
        spanning[u][v] = distance[v];
        spanning[v][u] = 

Certainly! Continuing from where we left off:

        spanning[u][v] = distance[v];
        spanning[v][u] = distance[v];

Lines 44-45: These lines add the edge between u and v with v being the newly added vertex to the minimum spanning tree (MST) being built in the spanning matrix. The cost of these edges is the minimum distance found in the previous step. The graph being undirected means the edge is bidirectional (u-v and v-u have the same weight), so the cost is updated for both [u][v] and [v][u].

        visited[v] = 1;
        no_of_edges--;

Lines 47-48: The vertex v is now marked as visited, meaning it's included in the MST, and hence the count of no_of_edges is decremented as one edge has been added to the MST.

        // Update the distance[] array
        for (i = 1; i < n; i++) {
            if (visited[i] == 0 && cost[i][v] < distance[i]) {
                distance[i] = cost[i][v];
                from[i] = v;
            }
        }
        min_cost = min_cost + cost[u][v];
    }

Lines 50-56: Update the distance[] array for all unvisited vertices. If the cost of adding any vertex i to the tree through v is less than the current known distance, then the value is updated. Essentially this step is about finding if there's a cheaper connection to the MST available for each vertex. Lastly, the min_cost is updated by adding the cost of the edge just added to the MST (the edge between u and v).

    return (min_cost);
}

Lines 58-59: At the end of the function, after the loop has added all necessary edges to form the MST, the total cost of the MST min_cost is returned.

void print() {
    int i, j, cost = 0;
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++) {
            if (spanning[i][j] != 0) {
                printf("\n%d-%d:\t%d", i, j, spanning[i][j]);
                cost = cost + spanning[i][j];
            }
        }
    }
    printf("\n\nCost of the spanning tree=%d", cost/2);
}

Lines 61-71: The print function iterates over the spanning matrix and prints out the edges that make up the MST and calculates the cost of the spanning tree. Since the MST is represented in a symmetric matrix (because the graph is undirected), every edge is counted twice, hence the cost is divided by 2 before it is printed to give the correct weight of the MST.

int main() {
    int i, j, total_cost;
    printf("Enter no. of vertices:");
    scanf("%d", &n);

73-77: Start of the main function. First, it gets an integer input from the user for the number of vertices and stores it in n.

    printf("\nEnter the adjacency matrix:\n");
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++) {
            scanf("%d", &G[i][j]);
        }
    }

79-85: Next, the user is asked to input the adjacency matrix of the graph, filling up the global matrix G with the weights of the edges between vertices. Zeroes in this matrix represent no connectivity.

    total_cost = prims();
    print();
    printf("\n");
    return 0;
}

87-92: Calls the prims function and store the return value—the minimum cost of the spanning tree—in total_cost. The print function is called to output the edges included in the MST and its cost. Finally, the main function returns 0 indicating successful execution of the program.

That concludes the line-by-line explanation of the code. The code itself does not include the i variable in the for loops (as in for (i = 0; i < n; i++)), and the comment in the print function is incorrect as // Insert the in spanning tree should say // Insert the edge in spanning tree. These seem to be transcription errors. For the code to compile and run correctly, these errors would need to be fixed.