{"paper":{"title":"Counterexamples to a Conjecture on Laplacian Ratios of Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"This paper constructs infinite families of trees whose Laplacian ratio exceeds the value conjectured to be maximal.","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Priyanshu Pant","submitted_at":"2026-05-13T22:55:10Z","abstract_excerpt":"For a graph \\(G\\) with no isolated vertices, its Laplacian ratio is defined as \\[ \\pi(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 \\(\\pi(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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give infinite families of counterexamples to their conjecture.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The explicit trees in the infinite families satisfy π(T) larger than the conjectured maximum, which rests on correct computation of the permanent of L(T) and the degree product for those trees.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Infinite families of trees are shown to have strictly larger Laplacian ratios than those allowed by the conjecture of Wu, Dong and Lai, disproving it.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"This paper constructs infinite families of trees whose Laplacian ratio exceeds the value conjectured to be maximal.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"62c625a80341b2ec89e44ef8b2bffcfd0d4791ce659640597530ceb1021dbed6"},"source":{"id":"2605.14176","kind":"arxiv","version":1},"verdict":{"id":"81f0c114-0b9f-4fb9-9108-62343b6b9369","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:53:05.409645Z","strongest_claim":"We give infinite families of counterexamples to their conjecture.","one_line_summary":"Infinite families of trees are shown to have strictly larger Laplacian ratios than those allowed by the conjecture of Wu, Dong and Lai, disproving it.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The explicit trees in the infinite families satisfy π(T) larger than the conjectured maximum, which rests on correct computation of the permanent of L(T) and the degree product for those trees.","pith_extraction_headline":"This paper constructs infinite families of trees whose Laplacian ratio exceeds the value conjectured to be maximal."},"references":{"count":2,"sample":[{"doi":"10.1016/0012-365x(84)90127-4","year":1984,"title":"R. A. Brualdi and J. L. Goldwasser. Permanent of the Laplacian matrix of trees and bipartite graphs.Discrete Mathematics, 48:1–21, 1984.doi:10.1016/0012-365X(84)90127-4","work_id":"716ca745-510d-4f29-8b2f-260072e18f41","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.dam.2025.04.047","year":2025,"title":"T. Wu, X. Dong, and H.-J. Lai. Two problems on Laplacian ratios of trees.Discrete Applied Mathematics, 372:224–236, 2025.doi:10.1016/j.dam.2025.04.047. 8","work_id":"9889c794-0cfe-4470-bb6b-7e9a646d28a2","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":2,"snapshot_sha256":"171bdcd6737d42ef92302453be55e5ec2fedad10d84d39e9b3d9eb422c402d5f","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}