Extremizing antiregular graphs by modifying total $σ$-irregularity
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917827707928576 |
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| author | Knor, Martin Škrekovski, Riste Filipovski, Slobodan Dimitrov, Darko |
| author_facet | Knor, Martin Škrekovski, Riste Filipovski, Slobodan Dimitrov, Darko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremizing antiregular graphs by modifying total $σ$-irregularity Knor, Martin Škrekovski, Riste Filipovski, Slobodan Dimitrov, Darko Combinatorics 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. |
| title | Extremizing antiregular graphs by modifying total $σ$-irregularity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.01530 |