{"paper":{"title":"Equivalent characterizations of the spectra of graphs and applications to measures of distance-regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J. F\\`abrega, M.A. Fiol, V. Diego","submitted_at":"2016-07-30T08:27:45Z","abstract_excerpt":"As it is well known, the spectrum $ {\\rm sp\\,} \\Gamma$ (of the adjacency matrix $A$) of a graph $\\Gamma$, with $d$ distinct eigenvalues other than its spectral radius $\\lambda_0$, usually provides a lot of information about the structure of $G$. Moreover, from ${\\rm sp\\,}\\Gamma$ we can define the so-called predistance polynomials $p_0,\\ldots,p_d\\in {\\mathbb R}_d[x]$, with ${\\rm dgr\\,} p_i=i$, $i=0,\\ldots,d$, which are orthogonal with respect to the scalar product $\\langle f, g\\rangle_{\\Gamma} =\\frac{1}{n}{\\rm tr\\,}(f(A)g(A))$ and normalized in such a way that $\\|p_i\\|_{\\Gamma}^2=p_i(\\lambda_0)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00091","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"}