{"paper":{"title":"The surjectivity of the combinatorial Laplacian on infinite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AP","authors_text":"Jozef Dodziuk, Michel Coornaert, Tullio Ceccherini-Silberstein","submitted_at":"2011-03-25T04:05:10Z","abstract_excerpt":"Given a connected locally finite simplicial graph $ G$ with vertex set $V$, the combinatorial Laplacian $\\Delta_G \\colon \\R^V \\to \\R^V$ is defined on the space of all real-valued functions on $V$. We prove that $\\Delta_G$ is surjective if $G$ is infinite."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4901","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"}