{"paper":{"title":"Vertex weighted Laplacian graph energy and other topological indices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"H. Panahbar, Reza Sharafdini","submitted_at":"2016-09-06T08:11:56Z","abstract_excerpt":"Let $G$ be a graph with a vertex weight $\\omega$ and the vertices $v_1,\\ldots,v_n$. The Laplacian matrix of $G$ with respect to $\\omega$ is defined as\n  $L_\\omega(G)=\\mathrm{diag}(\\omega(v_1),\\cdots,\\omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $\\mu_1,\\cdots,\\mu_n$ be eigenvalues of $L_\\omega(G)$. Then the Laplacian energy of $G$ with respect to $\\omega$ defined as $LE_\\omega (G)=\\sum_{i=1}^n\\big|\\mu_i - \\bar{\\omega}\\big|$, where $\\bar{\\omega}$ is the average of $\\omega$, i.e., $\\bar{\\omega}=\\dfrac{\\sum_{i=1}^{n}\\omega(v_i)}{n}$. In this paper we consider several natural"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01425","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"}