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For any $i,j $ vertices of $G$, we define $D_{\\{i,j\\}} ({\\cal G})$ to be the minimum of the weights of the simple paths of $G$ joining $i$ and $j$. The $D_{\\{i,j\\}} ({\\cal G})$ are called $2$-weights of ${\\cal G}$.\n  Let $\\{m_I\\}_{I \\in {\\{1,...,n\\} \\choose 2}}$ and $\\{M_I\\}_{I \\in {\\{1,...,n\\"},"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":"1409.3863","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-12T21:02:00Z","cross_cats_sorted":[],"title_canon_sha256":"f836f2bf246571681bdb9a446963b390cc93e2234e5c34516d98518a26e96998","abstract_canon_sha256":"2e837915d10fee0cd1754d1678037fc8ccd6c9cea44a9c99897d44eb70e62316"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:03.761667Z","signature_b64":"iI1sEfXYN/C/7VNej44/PocHIiv6hbYCWW7NVNjNwz9Xea+xai7plgYSMJeb9NXdcrY53FJXOomclGbq3czFDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d3b75872625a7e9db3479a6a8f4b160a2a058b4b223fa7ad02fb649be848766","last_reissued_at":"2026-05-18T02:31:03.761036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:03.761036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weighted graphs with distances in given ranges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Elena Rubei","submitted_at":"2014-09-12T21:02:00Z","abstract_excerpt":"Let ${\\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. 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