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Entropy and the complexity of graphs: I. An index of the relative complexity of a graph

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Abstract

The structural information content (Rashevsky, 1955; Trucco 1956a, b)I g (X) of a graphX is defined as the entropy of the finite probability scheme constructed from the orbits of its automorphism groupG(X). The behavior ofI g on various graph operations—complement, sum, join, cartesian product and composition, is examined. The principal result of the paper is the characterization of a class of graph product operations on whichI g is semi-additive. That is to say, conditions are found for binary operations o and ∇ defined on graphs and groups, respectively, which are sufficient to insure thatI g (X o Y)=I g (X)+I g (Y)−H XY , whereH XY is a certain conditional entropy defined relative to the orbits ofG(X o Y) andG(X) ∇G(Y).

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Authors and Affiliations

  1. Mental Health Research Institute, The University of Michigan, Ann Arbor, Michigan

    Abbe Mowshowitz

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  1. Abbe Mowshowitz
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Mowshowitz, A. Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bulletin of Mathematical Biophysics 30, 175–204 (1968). https://doi.org/10.1007/BF02476948

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