Java Program to Find Laplacian Matrix of an Undirected Graph

The Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. To find the Laplacian matrix first, find adjacency matrix and degree matrix of a graph as the formula for the Laplacian matrix is as follows:

Laplacian matrix = Degree matrix – Adjacency matrix

Example

Laplacian Matrix: 
 2  -1   0   0  -1   0  
-1   3  -1   0  -1   0  
 0  -1   2  -1   0   0  
 0   0  -1   3  -1  -1  
-1  -1   0  -1   3   0  
 0   0   0  -1   0   1

Basic definitions:

1. Adjacency matrix: Value can be either 0 or 1 according to graph vertices are connected to each other. 
2. Degree matrix: Number of vertices adjacent to a vertex.

Algorithm:

1. Add appropriate edges of an undirected graph.
2. Reading adjacency and degree matrix of graph
3. Compute the Laplacian matrix with the formula.

Below is the implementation of the above approach:

Java

// Java program to find laplacian
// matrix of an undirected graph
class Graph {
    class Edge {
        int src, dest;
    }
    int vertices, edges;
    Edge[] edge;

    Graph(int vertices, int edges)
    {
        this.vertices = vertices;
        this.edges = edges;
        edge = new Edge[edges];
        for (int i = 0; i < edges; i++) {
            edge[i] = new Edge();
        }
    }
    public static void main(String[] args)
    {
        int i, j;
        int numberOfVertices = 6;
        int numberOfEdges = 7;
        int[][] adjacency_matrix
            = new int[numberOfEdges][numberOfEdges];
        int[][] degree_matrix
            = new int[numberOfEdges][numberOfEdges];
        int[][] laplacian_matrix
            = new int[numberOfEdges][numberOfEdges];
        Graph g
            = new Graph(numberOfVertices, numberOfEdges);

        // Adding edges with source and destination
        // edge 1--2
        g.edge[0].src = 1;
        g.edge[0].dest = 2;

        // edge 1--5
        g.edge[1].src = 1;
        g.edge[1].dest = 5;

        // edge 2--3
        g.edge[2].src = 2;
        g.edge[2].dest = 3;

        // edge 2--5
        g.edge[3].src = 2;
        g.edge[3].dest = 5;

        // edge 3--4
        g.edge[4].src = 3;
        g.edge[4].dest = 4;

        // edge 4--6
        g.edge[5].src = 4;
        g.edge[5].dest = 6;

        // edge 5--4
        g.edge[6].src = 5;
        g.edge[6].dest = 4;

        // Adjacency Matrix
        for (i = 0; i < numberOfEdges; i++) {
            for (j = 0; j < numberOfEdges; j++) {
                adjacency_matrix[g.edge[i].src]
                                [g.edge[i].dest]
                    = 1;
                adjacency_matrix[g.edge[i].dest]
                                [g.edge[i].src]
                    = 1;
            }
        }
        System.out.println("Adjacency matrix : ");
        for (i = 1; i < adjacency_matrix.length; i++) {
            for (j = 1; j < adjacency_matrix[i].length;
                 j++) {
                System.out.print(adjacency_matrix[i][j]
                                 + " ");
            }
            System.out.println();
        }
        System.out.println("\n");

        // Degree Matrix
        for (i = 0; i < numberOfEdges; i++) {
            for (j = 0; j < numberOfEdges; j++) {
                degree_matrix[i][i]
                    += adjacency_matrix[i][j];
            }
        }
        System.out.println("Degree matrix : ");
        for (i = 1; i < degree_matrix.length; i++) {
            for (j = 1; j < degree_matrix[i].length; j++) {
                System.out.print(degree_matrix[i][j] + " ");
            }
            System.out.println();
        }
        System.out.println("\n");

        // Laplacian Matrix
        System.out.println("Laplacian matrix : ");
        for (i = 0; i < numberOfEdges; i++) {
            for (j = 0; j < numberOfEdges; j++) {
                laplacian_matrix[i][j]
                    = degree_matrix[i][j]
                      - adjacency_matrix[i][j];
            }
        }
        for (i = 1; i < laplacian_matrix.length; i++) {
            for (j = 1; j < laplacian_matrix[i].length;
                 j++) {
                System.out.printf("%2d",
                                  laplacian_matrix[i][j]);
                System.out.printf("%s", "  ");
            }
            System.out.println();
        }
    }
}

Output

Adjacency matrix : 
0 1 0 0 1 0 
1 0 1 0 1 0 
0 1 0 1 0 0 
0 0 1 0 1 1 
1 1 0 1 0 0 
0 0 0 1 0 0 


Degree matrix : 
2 0 0 0 0 0 
0 3 0 0 0 0 
0 0 2 0 0 0 
0 0 0 3 0 0 
0 0 0 0 3 0 
0 0 0 0 0 1 


Laplacian matrix : 
 2  -1   0   0  -1   0  
-1   3  -1   0  -1   0  
 0  -1   2  -1   0   0  
 0   0  -1   3  -1  -1  
-1  -1   0  -1   3   0  
 0   0   0  -1   0   1