{"paper":{"title":"The Automorphism Group of the Reduced Complete-Empty $X-$Join of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Adel Tadayyonfar, Ali Reza Ashrafi","submitted_at":"2017-09-03T12:01:10Z","abstract_excerpt":"Suppose $X$ is a simple graph. The $X-$join $\\Gamma$ of a set of complete or empty graphs $\\{X_x \\}_{x \\in V(X)}$ is a simple graph with the following vertex and edge sets: \\begin{eqnarray*} V(\\Gamma) &=& \\{(x,y) \\ | \\ x \\in V(X) \\ \\& \\ y \\in V(X_x) \\},\\\\ E(\\Gamma) &=& \\{(x,y)(x^\\prime,y^\\prime) \\ | \\ xx^\\prime \\in E(X) \\ or \\ else \\ x = x^\\prime \\ \\& \\ yy^\\prime \\in E(X_x)\\}. \\end{eqnarray*} The $X-$join graph $\\Gamma$ is called reduced if for vertices $x, y \\in V(X)$, $x \\ne y$, $N_X(x) \\setminus \\{ y\\} = N_X(y) \\setminus \\{ x\\}$ implies that $(i)$ if $xy \\not\\in E(X)$ then the graphs $X_x$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00706","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"}