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## Graphs and Matrices

This example shows an application of sparse matrices and explains the relationship between graphs and matrices.

A graph is a set of nodes with specified connections between them. An example is the connectivity graph of the Buckminster Fuller geodesic dome (also a soccer ball or a carbon-60 molecule).

In MATLAB®, the graph of the geodesic dome can be generated with the BUCKY function.

```% Define the variables.
[B,V] = bucky;
H = sparse(60,60);
k = 31:60;
H(k,k) = B(k,k);

% Visualize the variables.
gplot(B-H,V,'b-');
hold on
gplot(H,V,'r-');
hold off
axis off equal
```

A graph can be represented by its adjacency matrix.

To construct the adjacency matrix, the nodes are numbered 1 to N. Then element (i,j) of the matrix is set to 1 if node i is connected to node j, and 0 otherwise.

```% Define a matrix A.
A = [0 1 1 0 ; 1 0 0 1 ; 1 0 0 1 ; 0 1 1 0];

% Draw a picture showing the connected nodes.
cla
subplot(1,2,1);
gplot(A,[0 1;1 1;0 0;1 0],'.-');
text([-0.2, 1.2 -0.2, 1.2],[1.2, 1.2, -.2, -.2],('1234')', ...
'HorizontalAlignment','center')
axis([-1 2 -1 2],'off')

% Draw a picture showing the adjacency matrix.
subplot(1,2,2);
xtemp=repmat(1:4,1,4);
ytemp=reshape(repmat(1:4,4,1),16,1)';
text(xtemp-.5,ytemp-.5,char('0'+A(:)),'HorizontalAlignment','center');
line([.25 0 0 .25 NaN 3.75 4 4 3.75],[0 0 4 4 NaN 0 0 4 4])
axis off tight
```

Here are the nodes in one hemisphere of the bucky ball, numbered polygon by polygon.

```subplot(1,1,1);
gplot(B(1:30,1:30),V(1:30,:),'b-');
for j = 1:30,
text(V(j,1),V(j,2),int2str(j),'FontSize',10);
end
axis off equal
```

To visualize the adjacency matrix of this hemisphere, we use the SPY function to plot the silhouette of the nonzero elements.

Note that the matrix is symmetric, since if node i is connected to node j, then node j is connected to node i.

```spy(B(1:30,1:30))
title('spy(B(1:30,1:30))')
```

Now we extend our numbering scheme to the whole graph by reflecting the numbering of one hemisphere into the other.

```[B,V] = bucky;
H = sparse(60,60);
k = 31:60;
H(k,k) = B(k,k);
gplot(B-H,V,'b-');
hold on
gplot(H,V,'r-');
for j = 31:60
text(V(j,1),V(j,2),int2str(j), ...
'FontSize',10,'HorizontalAlignment','center');
end
hold off
axis off equal
```

Finally, here is a SPY plot of the final sparse matrix.

```spy(B)
title('spy(B)')
```

In many useful graphs, each node is connected to only a few other nodes. As a result, the adjacency matrices contain just a few nonzero entries per row.

This example has shown one place where SPARSE matrices come in handy.

```gplot(B-H,V,'b-');
axis off equal
hold on
gplot(H,V,'r-');
hold off
```

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