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In \\cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree $\\Delta= 5$ that are neith"},"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":"1106.1809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-06-09T13:43:40Z","cross_cats_sorted":[],"title_canon_sha256":"50001e726797d14500a48307b892ab2dadde5f65ccd7c2c7a6f676d632a9ad2b","abstract_canon_sha256":"f9b34e6c23948c04bc89ec271baa59ba9d67c6a94f25eff574c3e526f67ab804"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:44.017258Z","signature_b64":"J9JqlCq0SDDuBTddonWKi9XVKgbJDQzyf0sx/zgzcP+WI6FQR1onBV/iargVni7ltApkyFIfo4vkgs15IoAKCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cc5b3a5f2d3cb147027fc732b80259c9af0aff2cbf042337abe3f2b4502c8bd","last_reissued_at":"2026-05-18T02:21:44.016635Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:44.016635Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Zagreb Indices Equality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Darko Dimitrov, Hosam Abdo, Ivan Gutman","submitted_at":"2011-06-09T13:43:40Z","abstract_excerpt":"For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\\sum_{v\\in V}d(v)^2 $ and $M_2(G)=\\sum_{uv\\in E}d(u)d(v)$. In \\cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$ (also known as the Zagreb indices equality) if and only if $G$ is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). 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