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Sets of vertices with extremal energy

Neal Bushaw, Brent Cody, Chris Leffler

TL;DR

...

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.

Sets of vertices with extremal energy

TL;DR

...

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.
Paper Structure (8 sections, 21 theorems, 57 equations, 9 figures)

This paper contains 8 sections, 21 theorems, 57 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Minimizers on cycles
  3. Maximal evenness and J-representations
  4. Majorization
  5. Maximal evenness and majorization
  6. Complements of minimizers and maximizers
  7. Maximizers of the Wiener index on paths and cycles
  8. Future Directions

Key Result

Theorem 2.3

Suppose $m$ and $n$ are integers with $1\leq m\leq n$. Then a set of vertices is maximally even in $C_n$ if and only if it is of the form $J^r_{n,m}$ for some integer $r$ with $0\leq r\leq n$.

Figures (9)

  • Figure 1: Examples of minimizers of $E_g$, where $g(r)=\frac{1}{r}$, in the Petersen graph (A) and the hypercube (B).
  • Figure 2: The shaded vertices indicate a global maximizer of $W$ (A), a local maximizer of $W$ that is not global (B), a global minimizer of $E_g$ (C) and a local minimizer of $E_g$ that is not global (D), where $g(r)=\frac{1}{r}$ (see Example \ref{['example_locnotglo']}).
  • Figure 3: Minimizers of $E_g$ on various Möbius ladders where $g(r)=\frac{1}{r}$.
  • Figure 4: The maximally even set $J^0_{12,5}$ is shown in black (A), a rotation of the complement of $J^0_{12,5}$ is written in standard notation as both a musical scale and rhythm (B) and $J^0_{12,5}$ as well as its complement can be visualized as the black keys and white keys, respectively, on a piano keyboard.
  • Figure 5: A pair $(a_i,a_j)$ with $\textrm{span}_A(a_i,a_j)=t=\left\lfloor \frac{m}{2}\right\rfloor$ and $d^*(a_i,a_j)>\left\lfloor \frac{n}{2}\right\rfloor$.
  • ...and 4 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 2.1: Clough and Douthett CloughDouthett
  • Definition 2.2: Clough and Douthett CloughDouthett
  • Theorem 2.3: Clough and Douthett CloughDouthett
  • Example 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • ...and 32 more