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By Richard S.H. Mah, Howard Brenner, Andreas Acrivos and James E. Bailey (Auth.)

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Extra resources for Chemical Process Structures and Information Flows

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4. Parallel edges are represented by identical columns. 5. Self-loops cannot be represented in an incidence matrix. 6. Permutation of rows (columns) correspond to the interchange of ver­ tex (edge) labels. 7. )^, die rows are linearly dependent and the rank of an incident matrix is less tiian or equal to Ν - 1 . The last characteristic is very important It can be readily shown that for a connected digraph the rank of its incidence matrix is always Ν - Í (Problem 2-36). We may, therefore, omit one row of M' without any loss of information.

The additional condition for two digraphs to be isomorphic is that the directions of the corresponding edges must be the same. For instance, for a given vertex set not all complete asymmetric digraphs are isomorphic to each other. In many applications it is often useful to refer to both the digraph G and the undirected graph corresponding to G. This dual representation allows us to deal more directiy witii properties which do not depend on directions as well as those which are clearly related to the directions.

2 Fig. 2-21 States of a 4-component system. Although the modeling assumptions (sharp cuts and no energy inte­ gration) are quite severe, this treatment is interesting in that it demonstrates structural similarity in apparentiy very different problems. 2-7. MATRIX REPRESENTATION OF DIGRAPHS AND GRAPHS For small numbers of vertices and edges pictorial representation of a graph is both convenient and instructive. But this representation rapidly becomes unmanageable as tiiese numbers increase. Practical problems which really require the use of graph-theoretic techniques are usually of high dimensions.

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