Pith Number

pith:LHBGF5RB

pith:2026:LHBGF5RB27TZVW3KK34Z547EYA

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On the Extremal Energy of Complex Unit Gain Dumbbell Graphs

Kevin Pereyra, Silin Huang

The characteristic polynomial of complex unit gain dumbbell graphs is expressed using matching polynomials of subgraphs, solving extremal energy problems except one case.

arxiv:2604.27785 v2 · 2026-04-30 · math.CO

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Claims

C1strongest claim

An explicit expression of its characteristic polynomial is derived in terms of the matching polynomials of some of its subgraphs. This is used to build two methods to solve the problem in different parity cases. ... The problems are solved for all parity cases except for the minimum energy problem when r,s are odd and ℓ is odd.

C2weakest assumption

The coefficient-comparison and integral-kernel methods suffice to locate the extrema once the characteristic polynomial is known, except in the all-odd minimum-energy case where the ordering of energies is not settled by the same analysis and requires separate treatment.

C3one line summary

Explicit characteristic polynomial for complex unit gain dumbbell graphs D_{r,s,ℓ} enables solving extremal energy problems in all parity cases except the minimum when r, s, and ℓ are all odd.

Receipt and verification

First computed	2026-05-27T01:05:55.461302Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

59c262f621d7e79adb6a56f99ef3e4c0034b52b2442ccd9c9c1348dd3b2729ab

Aliases

arxiv: 2604.27785 · arxiv_version: 2604.27785v2 · doi: 10.48550/arxiv.2604.27785 · pith_short_12: LHBGF5RB27TZ · pith_short_16: LHBGF5RB27TZVW3K · pith_short_8: LHBGF5RB

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LHBGF5RB27TZVW3KK34Z547EYA \
  | 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: 59c262f621d7e79adb6a56f99ef3e4c0034b52b2442ccd9c9c1348dd3b2729ab

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "979e4bd2d2535bc3375de25fc4ef08dd921e323ae83f50b293976c72386bec14",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-04-30T12:27:21Z",
    "title_canon_sha256": "b78652fe04bd774bc270b9e9df6f7f7b7644329f27b16f28376b156d76725cb3"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.27785",
    "kind": "arxiv",
    "version": 2
  }
}