# Connected graph given adjacency matrix

77 views (last 30 days)
imperial1991 on 29 May 2012
Answered: Hon Wah Yeung on 31 Jan 2021
Hi all, I'm working on a research project on graphical models involving a large dimension (large number of nodes). I'm just wondering, is there an existing efficient algorithm to determine whether the graph is connected or not given its adjacency matrix? I've tried looking but it seems that the existing ones are brute force algorithms which are costly. Does MATLAB have a built-in function? When I checked, it seems that none exists as of now.
Help would be greatly appreciated!

Wolfgang Schwanghart on 29 May 2012
You can use the function dmperm to see if a graph consists of one or several connected components. E.g. see the example here http://blogs.mathworks.com/steve/2007/03/20/connected-component-labeling-part-3/
HTH, W.
imperial1991 on 30 May 2012
Hi wolfgang, this is great! thanks a lot!

Christine Tobler on 22 Dec 2016
Edited: Christine Tobler on 22 Nov 2019
I realize this is an old question, but since it's still getting visits, I have a small addition. As of R2015b, the new graph and digraph classes have a method for computing connected components. To check whether a graph is connected based on its adjacency matrix A, use
g = digraph(A);
bins = conncomp(g, 'Type', 'weak');
isConnected = all(bins == 1);
The vector bins gives the bin number for each node of A. If a graph is connected, all nodes will be in one bin, which is checked using all(bins == 1). This is not necessarily faster than dmperm, but easier to read.
##### 2 CommentsShowHide 1 older comment
Christine Tobler on 22 Nov 2019
Thank you, yes, this was a typo. I've edited the original answer to fix this (although it took me two years to realize you had commented on this answer).

Andrei Bobrov on 29 May 2012
try grTheory - Graph Theory Toolbox by Sergii Iglin
imperial1991 on 30 May 2012
hi andrei, it seems like there's no function for checking whether a graph is connected or not. thanks anyway!

Hon Wah Yeung on 31 Jan 2021
This will work if you can at least load the matrix (meaning the matrix is not larger than the max possible array for Matlab) to Matlab.
function [idx,Group] = CheckConnected(E)
L=length(E);
T=find(E(:,1)~=0);
T=[T;1]';
M=sum(E(:,T),2);
T1=union(find(M~=0),T);
while length(T1)>length(T)
T=T1;
M=sum(E(:,T),2);
T1=union(find(M~=0),T);
end
if length(T)==L
idx=1;
Group=T;
else
idx=0;
Group=T;
end
end