# Properties of adjacency matrix

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Hey

following thought about the adjacency matrix of a graph.

Is it possible to distinguish from the adjacency matrix of a graph if the whole system of points is interconnected, or if there are 2 or more subsystems, whcih have connections inside each subsystem but the subsystems are not connected to each other.

Which properties does the adjacency matrix need to represent a interconnected system, or where do i have to seed the "ones" in the matrox to ensure the whole system is interconnected?

I havent figured it out on my own my some example systems, but maybe someone has the answer on this already.

many thanks in advance

##### 0 Comments

### Answers (4)

Bruno Luong
on 17 Jul 2020

Checkout conncomp

##### 3 Comments

Steven Lord
on 17 Jul 2020

Christine Tobler
on 27 Jul 2020

In terms of the adjacency matrix, a disconnected graph means that you can permute the rows and columns of this matrix in a way where the new matrix is block-diagonal with two or more blocks (the maximum number of diagonal blocks corresponds to the number of connected components).

If you want to compute this from scratch, you'll be better off using graph-style algorithms instead of matrix terminology, specifically a breadth-first search or depth-first search. These can be implemented in terms of the adjacency matrix, although it will be less efficient than the built-in used in the graph object. See https://en.wikipedia.org/wiki/Component_(graph_theory) which has some discussion of the algorithms involved.

##### 1 Comment

Bruno Luong
on 27 Jul 2020

Here is a quick and dirty check of graph connexity using adjacent matrix. But I agree why not using stock function that is much better implemented (there is also some code on the FEX using graph algo, that once I checked out - can't remember the name - but I ended up using my own on implementation)

% A is the adjacent matrix, assumed to be symmetric (undirect graph)

n = size(A,1);

x = zeros(n,1);

x(1) = 1;

c = 1; % sum(x)

while true

x = A*x > 0 | x;

s = sum(x);

if s == c

break

end

c = s;

end

if c == n

fprintf('Single component\n');

else

fprintf('Multiple components\n');

end

Bruno Luong
on 27 Jul 2020

Edited: Bruno Luong
on 27 Jul 2020

Here is a quick and dirty calculation of connexed component using adjacent matrix. It returns the same output as MATLAB conncomp function in BINS and BINSIZES.

% TMW example

s = [1 2 2 3 3 3 4 5 5 5 8 8 9];

t = [2 3 4 1 4 5 5 3 6 7 9 10 10];

G = digraph(s,t,[],20);

A = G.adjacency;

A = spones(A + A'); % no need to symmetrize for undirected graph

n = size(A,1);

bins = zeros(n,1);

k = 0;

while any(bins==0)

x = sparse(find(~bins,1), 1, true, n, 1, n);

y = x;

while true

y = A*y;

y = y > 0 & ~x;

if nnz(y) == 0

break

end

x = x | y;

end

k = k+1;

bins(x) = k;

end

% Formating output

nodes = (1:n)';

Comptable = table(nodes, bins);

CompSet = accumarray(bins, nodes, [], @(x) {sort(x)'});

binsizes = accumarray(bins, 1);

Display

plot(G,'Layout','layered')

k = length(CompSet);

fprintf('Graph has %d bins(s)\n', k);

Comptable

binsizes

for i=1:k

fprintf('Compnent %d: Nodes = %s\n', i, mat2str(CompSet{i}));

end

##### 0 Comments

Bruno Luong
on 27 Jul 2020

Edited: Bruno Luong
on 27 Jul 2020

Another way - most direct perhaps link to matrix property - is using this property of Laplacian matrix:

"The number of connected components in the graph is the dimension of the nullspace of the Laplacian and the algebraic multiplicity of the 0 eigenvalue." from https://en.wikipedia.org/wiki/Laplacian_matrix

Not sure how is the numerical stability (probably not very reliable).

% TMW example

s = [1 2 2 3 3 3 4 5 5 5 8 8 9];

t = [2 3 4 1 4 5 5 3 6 7 9 10 10];

G = graph(s,t);

A = G.adjacency;

Use Laplacian

D = diag(sum(A)); % degree matrix

L = D - A; % laplacian matrix

[Q,R,P] = qr(L);

nc = full(sum(abs(diag(R)) < eps)) % number of components

##### 1 Comment

Bruno Luong
on 28 Jul 2020

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