{"paper":{"title":"Adjacency Matrices of Configuration Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Labbate, M. Abreu, M. Funk, V. Napolitano","submitted_at":"2010-02-04T16:20:28Z","abstract_excerpt":"In 1960, Hoffman and Singleton \\cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (\\kappa - 1) I_n + J_n - A A^{\\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $\\kappa$, respectively. If $A$ is an incidence matrix of some configuration $\\cal C$ of type $n_\\kappa$, then the left-hand side $\\Theta(A):= (\\kappa - 1)I_n + J_n - A A^{\\rm T}$ is an adjacency matrix of the non--collinearity graph $\\Gamma$ of $\\cal C$. In certain situations, $\\Theta(A)$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1032","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}