public int numDistance2Neighbors(String vertex)
        {
                int index = getIndex(vertex);

                // Check if vertex is missing. If so, report an error
                if(index == -1)
                {
                        System.out.print("In numDistance2Neighbors: ");
                        System.out.print(vertex);
                        System.out.println(" is missing.");
                        return;
                }

                // get the indices of neighbors of index
                int[] neighbors = getNeighbors(index);

                // define a 2-d int array to store distance-2 neighbors
                int[][] distance2Neighbors = new int[neighbors.length][];

                // Get the neighbors of each neighbor of the vertex
                // and store them in distance2Neighbors array
                for(int i = 0; i < neighbors.length; i++)
                {
                        // store the neighbors of the ith neighbor
                        // into temp
                        int[] temp = getNeighbors(neighbors[i]);

                        // Copy the ith neighbor and the neighbors of the
                        // ith neighbor into temp
                        distance2Neighbors[i] = new int[temp.length+1];
                        distance2Neighbors[i][0] = neighbors[i];

                        for(int j = 0; j < temp.length; j++)
                                distance2Neighbors[i][j+1] = temp[j];
                }

		// count the number of distinct indices stored in
		// the 2-d array distance2Neighbors and return this value
                return countDistinct(distance2Neighbors);

        }

	
	// function that counts the number of distinct elements in the given 
	// 2-d array
	int countDistinct(int[][] distance2Neighbors)
	{

		// index to keep track of the number of distinct elements
		int count = 0;

		// Process each index in the 2-d array and check if it has
		// already appeared. If it has, then do not increment count;
		// otherwise, increment count.
		for(int r = 0; r < distance2Neighbors.length; r++)
			for(int c = 0; c < distance2Neighbors[r].length; c++)
				if(isDuplicate(distance2Neighbors, r, c))
					count++;

		return count;
	}

	// Determines if the element in the rth row, cth column already
	// appears in the array
	boolean isDuplicate(int[][] theArray, int r, int c)
	{

		// This nested loop scans elements in all previous rows
		// to see if there is an element identical to the one in the
		// rth row, cth column
		for(int i = 0; i < r; i++)
			for(int j = 0; j < distance2Neighbors[i].length; j++)
				if(distance2Neighbors[i][j] == distance2Neighbors[r][c])
					return false;

		return true;
	}