{"paper":{"title":"The weighted vertex PI index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Ili\\' c, Nikola Milosavaljevi\\' c","submitted_at":"2011-04-21T13:30:43Z","abstract_excerpt":"The vertex PI index is a distance--based molecular structure descriptor, that recently found numerous chemical applications. In order to increase diversity of this topological index for bipartite graphs, we introduce weighted version defined as $PI_w (G) = \\sum_{e = uv \\in E} (deg (u) + deg (v)) (n_u (e) + n_v (e))$, where $deg (u)$ denotes the vertex degree of $u$ and $n_u (e)$ denotes the number of vertices of $G$ whose distance to the vertex $u$ is smaller than the distance to the vertex $v$. We establish basic properties of $PI_w (G)$, and prove various lower and upper bounds. In particula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4259","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"}