Entropy and the complexity of graphs: IV. Entropy measures and graphical structure

Abstract

The structural information contentI _g (X) of a graphX was treated in detail in three previous papers (Mowshowitz 1968a, 1968b, 1968c). Those investigations ofI _g point up the desirability of defining and examining other entropy-like measures on graphs. To this end the chromatic information contentI _c (X) of a graphX is defined as the minimum entropy over all finite probability schemes constructed from chromatic decompositions having rank equal to the chromatic number ofX. Graph-theoretic results concerning chromatic number are used to establish basic properties ofI _c on arbitrary graphs. Moreover, the behavior ofI _c on certain special classes of graphs is examined. The peculiar structural characteristics of a graph on which the respective behaviors of the entropy-like measuresI _c andI _g depend are also discussed.