Sets of vertices with extremal energy

TL;DR

This paper introduces new notions of vertex set energy in graphs, generalizing classical indices, and characterizes extremal sets in cycles and paths, linking energy minimization to graph regularity.

Contribution

It generalizes existing results on vertex energy, provides new characterizations of extremal vertex sets, and links energy properties to graph regularity and structure.

Findings

Sets of minimal energy in cycles are maximally even sets.

Distance degree regular graphs have symmetric energy properties.

Characterizations of extremal sets in paths and cycles for maximal distance sums.

Abstract

We define various notions of energy of a set of vertices in a graph, which generalize two of the most widely studied graphical indices: the Wiener index and the Harary index. We provide a new proof of a result due to Douthett and Krantz, which says that for cycles, the sets of vertices which have minimal energy among all sets of the same size are precisely the maximally even sets, as defined in Clough and Douthett's work on music theory. Generalizing a theorem of Clough and Douthett, we prove that a finite, simple, connected graph is distance degree regular if and only if whenever a set of vertices has minimal energy, its complement also has minimal energy. We also provide several characterizations of sets of vertices in finite paths and cycles for which the sum of all pairwise distances between vertices in the set is maximal among all sets of the same size.