Is there a common / good way to represent graphs in Matlab and operate on them? I need to read in an image as a graph and use several algorithms of graph-theory like minimal cut.
Matlab has a limited functionality network analysis toolbox. Yet, I guess it covers what you might need: http://www.mathworks.com/help/bioinfo/network-analysis-and-visualization.html
Matlab is not exactly the best way to deal with graphs. However, by using sparse matrices and mex files, many of the operations can be efficiently performed. In that sense, I think the below 3rd parth toolboxes has some good computational capabilities:
http://www.mathworks.com/matlabcentral/fileexchange/4266-grtheory-graph-theory-toolbox http://www.mathworks.com/matlabcentral/fileexchange/10922-matlabbgl http://www.mathworks.com/matlabcentral/fileexchange/5355-toolbox-graph http://www.mathworks.com/matlabcentral/fileexchange/19218-matgraph