Extremal entropy for graphs with given size

Abstract

The first degree-based graph entropy of a graph is the Shannon entropy of its degree sequence. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based graph entropy are precisely the colex graphs.

Extremal values of degree-based entropies of bipartite graphs

“…Now one can note that I 1 = I 2 by implementing the following two equalities in Equation ( 1) and Equation (2). Having proven this particular case in Lemma 13, we continue with some observations from which we can conclude that the extremal tableaux will be as given in Section 3.…”

“…The interpretation of the entropy of a particular graph depends on knowing the extremal values. In this paper, we continue our work [2,3] on determining the extremal graphs (and thus extremal values) for the entropy among all graphs satisfying some natural restrictions. Minimizing (resp.…”

Resolution of Yan's conjecture on entropy of graphs

“…In this section, we prove the following theorem that gives the precise characterization of extremal graphs for h c (G) where c ≥ 1 is an integer (for c = 0, this was done in [2]).…”

“…Then by taking a vertex of minimum degree and relating h c (G) with h c (G\v), we perform the induction in Section 3. Besides a few exceptions, the extremal graphs turn out to be the star, contrary to the extremal graphs for h(G) when only the graph size is given, for which the extremal graphs are colex graphs, see [2]. The precise statement is formulated in Theorem 7.…”

“…In [2], we determined the minimum first degree-based entropy among all graphs with a given size. Here the extremal graphs are precisely the colex graphs.…”