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unmesh

Convert edge matrix to coordinate and Laplacian matrices

Syntax

[L,XY] = unmesh(E)

Description

[L,XY] = unmesh(E) returns the Laplacian matrix L and mesh vertex coordinate matrix XY for the M-by-4 edge matrix E. Each row of the edge matrix must contain the coordinates [x1 y1 x2 y2] of the edge endpoints.

Input Arguments

EM-by-4 edge matrix E.

Output Arguments

LLaplacian matrix representation of the graph.
XYMesh vertex coordinate matrix.

Examples

Take a simple example of a square with vertices at (1,1), (1,–1),(–1,–1), and (–1,1), where the connections between vertices are the four perpendicular edges of the square plus one diagonal connection between (–1, –1) and (1,1).

The edge matrix E for this graph is:

E=[1 1 1 -1;  % edge from 1 to 2
1 -1 -1 -1;   % edge from 2 to 3
-1 -1 -1 1;   % edge from 3 to 4
-1 -1 1 1;    % edge from 4 to 1
-1 1 1 1]     % edge from 3 to 1
Use unmesh to create the output matrices,
[A,XY]=unmesh(E);
4 vertices:
4/4
The Laplacian matrix is defined as

Lij={deg(vi)     if i=j1             if ij and vi is adjacent to vj0               otherwise

unmesh returns the Laplacian matrix L in sparse notation.

L

L =

   (1,1)        3
   (2,1)       -1
   (3,1)       -1
   (4,1)       -1
   (1,2)       -1
   (2,2)        2
   (4,2)       -1
   (1,3)       -1
   (3,3)        2
   (4,3)       -1
   (1,4)       -1
   (2,4)       -1
   (3,4)       -1
To see L in regular matrix notation, use the full command.
full(L)

ans =

     3    -1    -1    -1
    -1     2     0    -1
    -1     0     2    -1
    -1    -1    -1     3
The mesh coordinate matrix XY returns the coordinates of the corners of the square.
XY

XY =

    -1    -1
    -1     1
     1    -1
     1     1

See Also

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