An inverse result for Wang's theorem on extremal trees
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866916781391609856 |
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| author | Damnjanović, Ivan Ranđelović, Žarko |
| author_facet | Damnjanović, Ivan Ranđelović, Žarko |
| contents | Among all trees on $n$ vertices with a given degree sequence, how do we maximise or minimise the sum over all adjacent pairs of vertices $x$ and $y$ of $f(\mathrm{deg} x, \mathrm{deg} y)$? Here $f$ is a fixed symmetric function satisfying a 'monotonicity' condition that \[
f(x, a) + f(y, b) > f(y, a) + f(x, b) \quad \mbox{for any $x > y$ and $a > b$} . \] These functions arise naturally in several areas of graph theory, particularly chemical graph theory.
Wang showed that the so-called 'greedy' tree maximises this quantity, while an 'alternating greedy' tree minimises it. Our aim in this paper is to solve the inverse problem: we characterise precisely which trees are extremal for these two problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03048 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An inverse result for Wang's theorem on extremal trees Damnjanović, Ivan Ranđelović, Žarko Combinatorics 05C35, 05C05, 05C07, 05C92 Among all trees on $n$ vertices with a given degree sequence, how do we maximise or minimise the sum over all adjacent pairs of vertices $x$ and $y$ of $f(\mathrm{deg} x, \mathrm{deg} y)$? Here $f$ is a fixed symmetric function satisfying a 'monotonicity' condition that \[ f(x, a) + f(y, b) > f(y, a) + f(x, b) \quad \mbox{for any $x > y$ and $a > b$} . \] These functions arise naturally in several areas of graph theory, particularly chemical graph theory. Wang showed that the so-called 'greedy' tree maximises this quantity, while an 'alternating greedy' tree minimises it. Our aim in this paper is to solve the inverse problem: we characterise precisely which trees are extremal for these two problems. |
| title | An inverse result for Wang's theorem on extremal trees |
| topic | Combinatorics 05C35, 05C05, 05C07, 05C92 |
| url | https://arxiv.org/abs/2212.03048 |