{"paper":{"title":"Path-Minimality for Positive $p$-Energies, Laplacian-Type Spectra, and Line Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Quanyu Tang, Yinchen Liu","submitted_at":"2026-06-30T00:15:02Z","abstract_excerpt":"We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $$\n  \\mathcal E_p^+(G)\\ge \\mathcal E_p^+(P_n), $$ where $P_n$ is the path on $n$ vertices, in three settings: connected bipartite graphs for every real $p\\ge2$, all connected graphs for every odd integer $p\\ge3$, and all connected graphs for $p=4$. Second, using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and threshold"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30996","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30996/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}