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This question is also relevant in statistical mechanics because $W(\\{G\\},q)=\\exp(S_0/k_B)$, where $S_0$ is the ground state entropy of the $q$-state Potts antiferromagnet on the lattice graph $\\{G\\}$, and the analyti"},"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":"cond-mat/9707096","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cond-mat.stat-mech","submitted_at":"1997-07-09T19:36:10Z","cross_cats_sorted":["hep-lat","math.CO"],"title_canon_sha256":"bb335868ed18d6470587dec68796e5f6664c2ea227071b357eaa6c301c9614ac","abstract_canon_sha256":"b46d50b2ead84a82c4fa32fe41ad051f050dbbf6de3a239ab168e24b3b7017e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T16:04:04.629771Z","signature_b64":"VgTf6tvMOfkzcGwSsW2CMFGac+7Pg4wp4hCSuypZLsPbjqCKCUGIbSygP9EteW9/G/Bm6kWdD3ybmgzhjXOIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0b55fd3db26b8c3989f905a578cf17dddf60ed0e8e49eae83a2d827049d24e6","last_reissued_at":"2026-07-04T16:04:04.629439Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T16:04:04.629439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0","license":"","headline":"","cross_cats":["hep-lat","math.CO"],"primary_cat":"cond-mat.stat-mech","authors_text":"Robert Shrock, Shan-Ho Tsai (Institute for Theoretical Physics, State University of New York at Stony Brook)","submitted_at":"1997-07-09T19:36:10Z","abstract_excerpt":"Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\\{G\\},q) = \\lim_{n \\to \\infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the following: is $W_r(\\{G\\},q) = q^{-1}W(\\{G\\},q)$ analytic or not at the origin of the $1/q$ plane? 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