Extremal values of degree-based entropies of bipartite graphs

TL;DR

This paper characterizes bipartite graphs that minimize or maximize degree-based entropy for given size and order, using Young tableaux, revealing extremal structures and their connections to number theory.

Contribution

It introduces a novel approach using Young tableaux to identify extremal bipartite graphs for degree-based entropy, including both minimal and maximal cases.

Findings

Complete bipartite graphs minimize entropy for given size and order.

Nearly complete bipartite graphs are also extremal.

The methods extend to other degree-based indices.

Abstract

We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young tableaux, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory, but they are easy to find for specific values of the order $n$ and size $m$. We also give a direct argument to characterize the graphs maximizing the entropy given order and size. We indicate that some of our ideas extend to other degree-based topological indices as well.