{"paper":{"title":"$\\textbf{k}$-neighborhood ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Leila Sharifan, Somayeh Moradi","submitted_at":"2026-06-06T13:18:09Z","abstract_excerpt":"In this paper, we introduce and investigate the $\\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\\textbf{k}=(k_1,\\ldots,k_n)\\in \\mathbb{N}^n$ satisfying $1\\leq k_i\\leq \\textrm{deg}_G(i)+1$ for all $i$, the $\\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\\textrm{NI}_{\\textbf{k}}(G)=\\sum_{i=1}^n\\, (\\textbf{x}_W:\\, W\\subseteq N_G[i],\\, |W|=k_i)$$ of $S$, where $\\textbf{x}_W="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08159","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08159/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}