{"paper":{"title":"On the Extremal Energy of Complex Unit Gain Dumbbell Graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"The characteristic polynomial of complex unit gain dumbbell graphs is expressed using matching polynomials of subgraphs, solving extremal energy problems except one case.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin Pereyra, Silin Huang","submitted_at":"2026-04-30T12:27:21Z","abstract_excerpt":"We study the extremal energy problem for complex unit gain graphs whose underlying graph is the dumbbell graph $D_{r,s,\\ell}$. Using switching equivalence, we reduce the spectrum to the real parts of the two cycle gains and obtain an explicit expression of the characteristic polynomial in terms of matching polynomials of natural subgraphs. For the bipartite case, we determine the extremal gain assignments by coefficient comparison. For the non-bipartite cases, we analyze the Coulson integral kernels. Finally, the maximum-energy conditions are determined in all cases, while the minimum-energy c"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The characteristic polynomial of complex unit gain dumbbell graphs is expressed using matching polynomials of subgraphs, solving extremal energy problems except one case.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9cb743fc7db7182e2f7ee0a3578c964cec414664ed3519239c0698528385d7d0"},"source":{"id":"2604.27785","kind":"arxiv","version":2},"verdict":{"id":"26140b08-6d90-4537-a3cf-8fea35067a73","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T06:28:38.698856Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"The characteristic polynomial of complex unit gain dumbbell graphs is expressed using matching polynomials of subgraphs, solving extremal energy problems except one case."},"integrity":{"clean":false,"summary":{"advisory":1,"critical":0,"by_detector":{"doi_compliance":{"total":1,"advisory":1,"critical":0,"informational":0}},"informational":0},"endpoint":"/pith/2604.27785/integrity.json","findings":[{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. 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