{"paper":{"title":"Properties of the Dot Product Graph of a Commutative Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Mohsen Mollahajiaghaei","submitted_at":"2016-01-19T18:50:43Z","abstract_excerpt":"Let $R$ be a commutative ring with identity and $n\\geq1$ be an integer. Let $R^{n}=R\\times\\cdots\\times R~(n~times)$. The \\textit{total dot product} graph, denoted by $TD(R,n)$ is a simple graph with elements of $R^{n}-\\{(0,0,\\ldots,0)\\}$ as vertices, and two distinct vertices $\\mathbf{x}$ and $\\mathbf{y}$ are adjacent if and only if $\\mathbf{x} \\cdot \\mathbf{y}=0\\in R$, where $\\mathbf{x} \\cdot \\mathbf{y}$ denotes the dot product of $\\mathbf{x}$ and $\\mathbf{y}$. In this paper, we find the structure of $TD(R\\times S,n)$ with respect to the structure of $TD(R,n)$ and $TD(S,n)$. In addition, we f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05034","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"}