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Extremizing antiregular graphs by modifying total $σ$-irregularity

Martin Knor, Riste Škrekovski, Slobodan Filipovski, Darko Dimitrov

Abstract

The total $σ$-irregularity is given by $ σ_t(G) = \sum_{\{u,v\} \subseteq V(G)} \left(d_G(u) - d_G(v)\right)^2, $ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing $σ_{t}$-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to $ \IR(G)= \sum_{\{u,v\} \subseteq V(G)} |d_G(u)-d_G(v)|^{f(n)}, $ where $n=|V(G)|$ and $f(n)>0$. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.

Extremizing antiregular graphs by modifying total $σ$-irregularity

Abstract

The total -irregularity is given by where indicates the degree of a vertex within the graph . It is known that the graphs maximizing -irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to where and . We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.

Paper Structure

This paper contains 5 sections, 6 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. Antiregular extension of $\sigma_t$-irregularity
  3. General graphs
  4. Trees
  5. Chemical graphs

Key Result

Theorem 1

For any integer $n \geq 1$ and non-increasing sequence $d_1 \geq d_2 \geq \dots \geq d_n$, there exists a graph with $n$ vertices having the respective degrees $d_1, d_2, \dots, d_n$ if and only if two conditions are satisfied:

Figures (1)

  • Figure 1: Illustration of two distinct graphs, $G_1$ and $G_2$, which share the degree sequence $1, 1, 1, 2, 2, 2, 3$. Despite their differing irregularity values (${\rm irr}(G_1)=10$ and ${\rm irr}(G_2)=8$), they exhibit identical total irregularity (${\rm irr}_t(G_1)={\rm irr}_t(G_2)=22$).

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 3
  • Theorem 4
  • proof
  • Theorem 7
  • proof
  • Theorem 9
  • proof
  • ...and 2 more