Trees with maximum $σ$-irregularity under a prescribed maximum degree 6
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866915767356751872 |
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| author | Bašić, Milan |
| author_facet | Bašić, Milan |
| contents | The sigma-irregularity index $σ(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $σ(T)$ within the class of trees of fixed order $n$ and bounded maximum degree $Δ= 6$. Using a penalty-function framework combined with handshake identities and congruence arguments, we determine the exact maximum value of $σ(T)$ for every residue class of $n$ modulo $6$, showing that the possible minimum values of the penalty function are $0, 10, 20, 22, 30,$ and $40$. For each case, we provide a complete characterization of all maximizing trees in terms of degree counts and edge multiplicities. In five of the six residue classes, all extremal trees contain only vertices of degrees $1, 2,$ and $6$, while for $n \equiv 3 \pmod{6}$ an additional exceptional family arises involving vertices of degree $3$. These results extend earlier work on sigma-irregularity for smaller degree bounds and illustrate the rapidly growing combinatorial complexity of the problem as the maximum degree increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01262 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Trees with maximum $σ$-irregularity under a prescribed maximum degree 6 Bašić, Milan Combinatorics Discrete Mathematics The sigma-irregularity index $σ(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $σ(T)$ within the class of trees of fixed order $n$ and bounded maximum degree $Δ= 6$. Using a penalty-function framework combined with handshake identities and congruence arguments, we determine the exact maximum value of $σ(T)$ for every residue class of $n$ modulo $6$, showing that the possible minimum values of the penalty function are $0, 10, 20, 22, 30,$ and $40$. For each case, we provide a complete characterization of all maximizing trees in terms of degree counts and edge multiplicities. In five of the six residue classes, all extremal trees contain only vertices of degrees $1, 2,$ and $6$, while for $n \equiv 3 \pmod{6}$ an additional exceptional family arises involving vertices of degree $3$. These results extend earlier work on sigma-irregularity for smaller degree bounds and illustrate the rapidly growing combinatorial complexity of the problem as the maximum degree increases. |
| title | Trees with maximum $σ$-irregularity under a prescribed maximum degree 6 |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2602.01262 |