Extremizing antiregular graphs by modifying total σ-irregularity

Martin Knor

doi:10.1016/j.amc.2024.129199

Extremizing antiregular graphs by modifying total σ-irregularity

2025-04-01

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DOI：10.1016/j.amc.2024.129199

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The total sigma-irregularity is given by sigma(t)(G)=& sum;({u,v}subset of V(G))(d(G)(u)-d(G)(v))(2), where d(G)(z) indicates the degree of a vertex z within the graph G. It is known that the graphs maximizing sigma(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 sigma(f(n))(t)(G)=& sum;({u,v}subset of 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

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