{"paper":{"title":"Moments in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C. Dalf\\'o, E. Garriga, M.A. Fiol","submitted_at":"2012-08-28T10:38:05Z","abstract_excerpt":"Let $G$ be a connected graph with vertex set $V$ and a {\\em weight function} $\\rho$ that assigns a nonnegative number to each of its vertices. Then, the {\\em $\\rho$-moment} of $G$ at vertex $u$ is defined to be $M_G^{\\rho}(u)=\\sum_{v\\in V} \\rho(v)\\dist (u,v) $, where $\\dist(\\cdot,\\cdot)$ stands for the distance function. Adding up all these numbers, we obtain the {\\em $\\rho$-moment of $G$}: $$ M_G^{\\rho}=\\sum_{u\\in V}M_G^{\\rho}(u)=1/2\\sum_{u,v\\in V}\\dist(u,v)[\\rho(u)+\\rho(v)]. $$ This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the {\\em Wiener "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5615","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":""},"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"}