{"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":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0009036174,0.0003224936,0.0005875739,0.0001130978,0.0001685623,0.00001372851,0.0004158045,0.0002955416,0.00007021544],"category_scores_gemma":[0.0001487601,0.0002351088,0.0003350072,0.0001973864,0.000220097,0.00008820268,0.0001622436,0.0001622526,0.000007791316],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00002828519,"about_ca_system_score_gemma":0.00003410821,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00003968347,"about_ca_topic_score_gemma":0.00004201946,"domain_scores_codex":[0.9981297,0.0002528538,0.000510146,0.0003746974,0.00008268032,0.0006498893],"domain_scores_gemma":[0.9982567,0.000511948,0.0003909514,0.0006180283,0.000086452,0.0001358751],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"bench_or_experimental","study_design_gemma":"bench_or_experimental","study_design_scores_codex":[0.0002278925,0.001108298,0.0004760757,0.00009466494,0.0005149049,0.00000449289,0.0004941633,2.371981e-7,0.6436481,0.3497628,0.002301937,0.001366448],"study_design_scores_gemma":[0.0008811026,0.0003045138,0.0006957148,0.00003079764,0.0001608594,0.00003627661,0.0004565895,0.00009646119,0.7934501,0.193798,0.009691183,0.0003984014],"study_design_candidate":"bench_or_experimental","study_design_consensus":"bench_or_experimental","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9477876,0.0005420474,0.04585665,0.0004445074,0.0002082177,0.001031843,0.003863725,0.0001021496,0.0001632276],"genre_scores_gemma":[0.9918578,0.00008451346,0.006670718,0.0003563572,0.0000754996,0.0002254837,0.0004239435,0.00005910293,0.0002466427],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.1559648,"threshold_uncertainty_score":0.9587458,"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."}}