# Program to find connected and non-isomorphic graphs

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Gamma1990 on 1 Oct 2021
Commented: Gamma1990 on 2 Oct 2021
Hello,
I want to determine the connectivity and isomorphism of the graph and save the connected unlabeled graph in the cell array.
In order to do this, I'm trying to make the program that determines connected and non-isomorphic graphs by defining adjacency matrix.
However, It doesn't seem to be working properly.
Although it can determine adjacency matrix that are connected graphs, it cannot determine adjacency matrix that are non-isomorphic graphs.
Here is the code.
% setting
nd = 4; % Number of nodes
eg = nd*(nd-1)/2; % Maximum number of edges
egs = dec2bin(0:2^eg-1)-'0'; % Combinations of edges
tri = find(triu(ones(nd), 1)); % Index of the upper triangle of the adjacency matrix
data = cell(1, 1); % Preallocation of the cell array
% Loop to find the first graph and save it in the cell array
index = 1;
for i = 1:2^eg % Repeat for several combinations of edges
adj(tri) = egs(i,:); % Assign combination to the upper triangle
G = graph(adj, 'upper'); % Defining a Graph
if all(~diff(conncomp(G))) % If it's connected
data{index, 1} = vertcat(adj); % Save the adjacency matrix in the cell array
index = index + 1;
if isempty(data{1, 1}) == 0 % Stop the loop if a connected graph is found
break
end
end
end
% A loop that determines whether a graph in the cell array and a graph that just calculated are non-isomorphism
% If they are non-isomorphism, save the graph's adjacency matrix that just calculated
for i = i+1:2^eg % Repeat for several combinations of edges
% Determine graph connectivity
adj(tri) = egs(i,:); % Assign combination to the upper triangle
G = graph(adj, 'upper'); % Defining a Graph
if not(all(~diff(conncomp(G)))) % If it's not connected
continue % Skip statements below and repeat next
else
% Loop to determine isomorphism
for i2 = 1:index-1
Gbefore = graph(adjbefore, 'upper'); % Access the adjacency matrix in the cell array
if not(isvector(isomorphism(G, Gbefore))) % If it's non-isomorphism
data{index, 1} = vertcat(adj); % Save the adjacency matrix in the cell array
index = index + 1;
break % Stop the loop if a non-isomorphic graph is found and repeat next
end
end
end
end
When I set the number of nodes 4, I can get 38 adjacency matrix.
However, according to (Number of Graphs on n unlabelled vertices (yorku.ca)), number of graphs on 4 unlabelled nodes is only 6.
How do I get this program to work properly?

Christine Tobler on 1 Oct 2021
In the last loop, you are checking if adj is isomorphic to any previous graph. If there is one graph that adj is not isomorphic to, it's added to the list.
However, you want to find out if adj is not isomorphic to any of the graphs in data, not just if it's not isomorphic to at least one of them.
Gamma1990 on 2 Oct 2021
Oops! I was careless.
The program worked well.

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