Counterexamples to a Conjecture on Laplacian Ratios of Trees
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866913126574718976 |
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| author | Pant, Priyanshu |
| author_facet | Pant, Priyanshu |
| contents | For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ π(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and \(\operatorname{per}\) denotes the permanent. Brualdi and Goldwasser asked for the maximum value of \(π(T)\) among trees \(T\) with a fixed number of vertices. Wu, Dong and Lai recently proposed a conjectural answer to this problem. We give infinite families of counterexamples to their conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14176 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counterexamples to a Conjecture on Laplacian Ratios of Trees Pant, Priyanshu Combinatorics Discrete Mathematics 05C05, 05C50, 15A15 For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ π(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and \(\operatorname{per}\) denotes the permanent. Brualdi and Goldwasser asked for the maximum value of \(π(T)\) among trees \(T\) with a fixed number of vertices. Wu, Dong and Lai recently proposed a conjectural answer to this problem. We give infinite families of counterexamples to their conjecture. |
| title | Counterexamples to a Conjecture on Laplacian Ratios of Trees |
| topic | Combinatorics Discrete Mathematics 05C05, 05C50, 15A15 |
| url | https://arxiv.org/abs/2605.14176 |