Graphs that Split Entropies

Abstract

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This concept is at the core of a new bounding technique for graph covering problems and has furnished the best known bounds for the problem of perfect hashing.The basis of the technique is the sub-additivity of graph entropy with respect to the union of graphs. The tightness of the bounds depends on whether or not we have equality rather than just sub-additivity. As a first step in this analysis, we are investigating whether for a given graph G the entropies of G and $\bar G$ add up to the entropy of the complete graph on the same vertex set, i.e., the entropy of the underlying probability distribution.We shall prove that for a bipartite graph G and an arbitrary probability distribution P on its vertex set the entropies of G and $\bar G$ add up to the entropy of P. Related problems will be discussed.The results have interesting connections with the Ford–Fulkerson theory of network...