{"id":"W2158093499","doi":"10.1186/1748-7188-7-14","title":"Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials","year":2012,"lang":"en","type":"article","venue":"Algorithms for Molecular Biology","topic":"Graph theory and applications","field":"Mathematics","cited_by":14,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Division of Mathematical Sciences; Pacific Institute for the Mathematical Sciences; National Science Foundation","keywords":"Adjacency matrix; Combinatorics; Mathematics; Eigenvalues and eigenvectors; Laplacian matrix; Adjacency list; Binary tree; Binary number; Characteristic polynomial; Weight-balanced tree; Invariant (physics); Tree (set theory); Spectrum (functional analysis); Discrete mathematics; Polynomial; Binary search tree; Graph; Arithmetic","routes":{"ca_aff":false,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.003741746,0.0003410794,0.001478297,0.002886085,0.002641998,0.003489768,0.001306125,0.001784655,0.005251603],"category_scores_gemma":[0.0509974,0.0005373827,0.0006710469,0.003471162,0.003990442,0.01021817,0.003395602,0.001811873,0.0009235246],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0008742698,"about_ca_system_score_gemma":0.0006819769,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0004867256,"about_ca_topic_score_gemma":0.0004423757,"domain_scores_codex":[0.9962389,0.0008604466,0.0002861624,0.001490611,0.0007641178,0.0003596022],"domain_scores_gemma":[0.9552104,0.02648826,0.006439702,0.008371538,0.00210386,0.001386249],"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":[0.0009913462,0.0001952778,0.09165532,0.0009905295,0.0002644544,0.001491295,0.00500681,0.01293032,0.02102794,0.7026323,0.009894415,0.1529201],"study_design_scores_gemma":[0.00004738683,0.00008336341,0.01566347,0.0001368319,0.00008960834,0.004138531,0.001652062,0.06150084,0.006261487,0.9004328,0.009904448,0.0000890873],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.7884545,0.001013264,0.1975854,0.001878809,0.0001337785,0.0000504631,0.0006537318,0.0004813903,0.009748627],"genre_scores_gemma":[0.979425,0.0001940952,0.01877108,0.0001938002,0.00007425165,0.0000387951,0.0004139614,0.00009537011,0.0007936735],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.005251603,"threshold_uncertainty_score":0.01978844,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.06327891369287991,"score_gpt":0.3557059889395285,"score_spread":0.2924270752466486,"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."}}