{"id":"W2963282263","doi":"10.1137/1.9781611975031.70","title":"A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank","year":2018,"lang":"en","type":"book-chapter","venue":"Society for Industrial and Applied Mathematics eBooks","topic":"Graph Theory and Algorithms","field":"Computer Science","cited_by":14,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Waterloo","funders":"","keywords":"Mathematics; Rank (graph theory); Hamiltonian (control theory); Combinatorics; Matrix (chemical analysis); Materials science; Mathematical optimization; Composite material","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.002452186,0.001908172,0.001447767,0.002197268,0.001659566,0.004451206,0.002866816,0.001627358,0.01430996],"category_scores_gemma":[0.02214872,0.0009321034,0.00144052,0.00260634,0.002836139,0.01325729,0.003178186,0.004582493,0.003144251],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.002302599,"about_ca_system_score_gemma":0.002110702,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001512257,"about_ca_topic_score_gemma":0.002472749,"domain_scores_codex":[0.9958212,0.0007102926,0.0001882836,0.001163686,0.001242563,0.0008739785],"domain_scores_gemma":[0.9811772,0.01202894,0.001231725,0.003475721,0.001253316,0.000833035],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.001186338,0.0004313738,0.008261227,0.0009833276,0.000187017,0.0004484628,0.001284527,0.05393346,0.03111698,0.7293742,0.0241655,0.1486276],"study_design_scores_gemma":[0.00009763132,0.0002737491,0.001583081,0.0001156915,0.0001108837,0.0004768073,0.000199551,0.222241,0.01303434,0.7507432,0.01100003,0.0001241021],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.2251312,0.002582229,0.6897565,0.006537878,0.0003896267,0.0003811945,0.003286681,0.003768115,0.06816658],"genre_scores_gemma":[0.8143554,0.001132095,0.1681974,0.001174384,0.0006858961,0.0003846039,0.001898017,0.0007661955,0.01140605],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.01430996,"threshold_uncertainty_score":0.04787165,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03601687008750448,"score_gpt":0.2479868785746317,"score_spread":0.2119700084871272,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}