Pith Number

pith:SBMYB2KD

pith:2026:SBMYB2KDCU5S34A6SMOLQX23DP

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Counterexamples to a Conjecture on Laplacian Ratios of Trees

Priyanshu Pant

This paper constructs infinite families of trees whose Laplacian ratio exceeds the value conjectured to be maximal.

arxiv:2605.14176 v1 · 2026-05-13 · math.CO · cs.DM

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Claims

C1strongest claim

We give infinite families of counterexamples to their conjecture.

C2weakest 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.

C3one 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.

References

2 extracted · 2 resolved · 0 Pith anchors

[1] 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 1984 · doi:10.1016/0012-365x(84)90127-4

[2] 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 2025 · doi:10.1016/j.dam.2025.04.047

Receipt and verification

First computed	2026-05-17T23:39:11.297784Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

905980e943153b2df01e931cb85f5b1bea7a1de632bd0b9c4a87b9f7ea3654ef

Aliases

arxiv: 2605.14176 · arxiv_version: 2605.14176v1 · doi: 10.48550/arxiv.2605.14176 · pith_short_12: SBMYB2KDCU5S · pith_short_16: SBMYB2KDCU5S34A6 · pith_short_8: SBMYB2KD

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/SBMYB2KDCU5S34A6SMOLQX23DP \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 905980e943153b2df01e931cb85f5b1bea7a1de632bd0b9c4a87b9f7ea3654ef

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6c1f240a1edf0961625189a58fbd82c0190df97e851f57a947e1bbe9ee8536a7",
    "cross_cats_sorted": [
      "cs.DM"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-13T22:55:10Z",
    "title_canon_sha256": "e641488c557e29bdf837a379c72b7463fbd3a0779167f75facac09ec3bfd7399"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14176",
    "kind": "arxiv",
    "version": 1
  }
}