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Extremizing antiregular graphs by modifying total σ-irregularity

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  • Knor, Martin
  • Škrekovski, Riste
  • Filipovski, Slobodan
  • Dimitrov, Darko

Abstract

The total σ-irregularity is given by σt(G)=∑{u,v}⊆V(G)(dG(u)−dG(v))2, where dG(z) indicates the degree of a vertex z within the graph G. It is known that the graphs maximizing σt-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 σtf(n)(G)=∑{u,v}⊆V(G)|dG(u)−dG(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.

Suggested Citation

  • Knor, Martin & Škrekovski, Riste & Filipovski, Slobodan & Dimitrov, Darko, 2025. "Extremizing antiregular graphs by modifying total σ-irregularity," Applied Mathematics and Computation, Elsevier, vol. 490(C).
    • Handle: RePEc:eee:apmaco:v:490:y:2025:i:c:s009630032400660x
    DOI: 10.1016/j.amc.2024.129199
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    1. Dimitrov, Darko & Stevanović, Dragan, 2023. "On the σt-irregularity and the inverse irregularity problem," Applied Mathematics and Computation, Elsevier, vol. 441(C).
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