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In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy summing all matrix elements g(x,y) is equal to the Euler characteristic X(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to X(G)."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.03369","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-07T23:47:46Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"8e32a04d5a9f3089e283fa8c371f2ff20f13e19d0600a98f4abf2bf2ac47c133","abstract_canon_sha256":"cabdfecb27b6b0905ee4c982bf2abf09f6d334cfbd5fbfc08309ae82695406bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:16.415002Z","signature_b64":"kh8OnlOnYiAKSfAQeZMkz9G2UdKG6xN+B8CbG5b1Q1ADTIpbhcYArkGHLPSnA6UhwBhpPGfhRE4py92OBKI3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5c8e8724853257cf524c7fce1d6e4b57b366bac1fb90540e6f0409b0717c0d2","last_reissued_at":"2026-05-17T23:41:16.414468Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:16.414468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The energy of a simplicial complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2019-07-07T23:47:46Z","abstract_excerpt":"A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. 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