{"id":"W4416785442","doi":"10.28924/2291-8639-23-2025-300","title":"Clique Numbers in Direct Sum Matrix Commuting Graphs: Structural Analysis and Optimal Bounds","year":2025,"lang":"","type":"article","venue":"International Journal of Analysis and Applications","topic":"Rings, Modules, and Algebras","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"","keywords":"Vertex (graph theory); Association scheme; Clique; Upper and lower bounds; Algebraic structure; Commutative ring; Ring (chemistry); Matrix (chemical analysis); Clique number","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"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":[0.001632453,0.000960379,0.0009508705,0.001776366,0.001486961,0.002675225,0.002496279,0.001321186,0.006524517],"category_scores_gemma":[0.01085002,0.0007541961,0.0008461427,0.001874363,0.00312801,0.005928542,0.002257925,0.002637781,0.0005829142],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.002101858,"about_ca_system_score_gemma":0.0008576293,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001254533,"about_ca_topic_score_gemma":0.002487713,"domain_scores_codex":[0.9987213,0.0003582193,0.00003566487,0.000382354,0.0002835473,0.0002188271],"domain_scores_gemma":[0.9893696,0.007780702,0.0007886245,0.0009778985,0.0004262518,0.0006568694],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"not_applicable","study_design_scores_codex":[0.0001413485,0.0001548497,0.001674759,0.0002000598,0.00003429508,0.0001155877,0.0003937934,0.04268213,0.005268516,0.9267422,0.003752319,0.0188401],"study_design_scores_gemma":[0.00002748445,0.00003922717,0.0006818689,0.00002883234,0.00002389745,0.00008759717,0.00009942174,0.2123111,0.002261929,0.7823861,0.002028854,0.00002366833],"study_design_candidate":"not_applicable","study_design_consensus":null,"genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.5052785,0.001738175,0.4365622,0.003781319,0.0001652012,0.0001438759,0.0009618531,0.0004624355,0.05090645],"genre_scores_gemma":[0.917015,0.0008739825,0.07515822,0.0003262334,0.0003205908,0.0002158252,0.0006119519,0.0002370437,0.005241223],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.006524517,"threshold_uncertainty_score":0.02182662,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.009948025447471087,"score_gpt":0.3286365604849578,"score_spread":0.3186885350374867,"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."}}