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arXiv · 2411.01530

Extremizing antiregular graphs by modifying total $\sigma$-irregularity

Martin Knor·Riste Škrekovski·Slobodan Filipovski·Darko Dimitrov
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Abstract

The total $\sigma$-irregularity is given by $ \sigma_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 $\sigma_{t}$-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 $ \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.

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Martin Knor, Riste Škrekovski, Slobodan Filipovski, Darko Dimitrov. 2024-11-03. Extremizing antiregular graphs by modifying total $\sigma$-irregularity. https://arxiv.org/abs/2411.01530

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