{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:RRT2LZXNTVU6FNWRGNAE3EUBXU","short_pith_number":"pith:RRT2LZXN","schema_version":"1.0","canonical_sha256":"8c67a5e6ed9d69e2b6d133404d9281bd360d0f0604e8a1c055cb482bcf2868b0","source":{"kind":"arxiv","id":"2606.08159","version":1},"attestation_state":"computed","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 $\\textbf{x}_W="},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.08159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-06-06T13:18:09Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5c2017a6c5e6d316626f4f81377a9949939ec7248ede09712a42795b929fc76b","abstract_canon_sha256":"12b9da789fdffcab7c146604878e40f57875ad1b21f71e5f7cf714a51570cc86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:28.556410Z","signature_b64":"1L0sNtbFx8JMMfJUpjmKo1iHqwfH7rKEeWs31pkDy5oSgQwKiFmyqgVvXdnkM37h6cewEmjMXKkVqsBoj2aqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c67a5e6ed9d69e2b6d133404d9281bd360d0f0604e8a1c055cb482bcf2868b0","last_reissued_at":"2026-06-09T01:05:28.555892Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:28.555892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.08159","created_at":"2026-06-09T01:05:28.555974+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08159v1","created_at":"2026-06-09T01:05:28.555974+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08159","created_at":"2026-06-09T01:05:28.555974+00:00"},{"alias_kind":"pith_short_12","alias_value":"RRT2LZXNTVU6","created_at":"2026-06-09T01:05:28.555974+00:00"},{"alias_kind":"pith_short_16","alias_value":"RRT2LZXNTVU6FNWR","created_at":"2026-06-09T01:05:28.555974+00:00"},{"alias_kind":"pith_short_8","alias_value":"RRT2LZXN","created_at":"2026-06-09T01:05:28.555974+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU","json":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU.json","graph_json":"https://pith.science/api/pith-number/RRT2LZXNTVU6FNWRGNAE3EUBXU/graph.json","events_json":"https://pith.science/api/pith-number/RRT2LZXNTVU6FNWRGNAE3EUBXU/events.json","paper":"https://pith.science/paper/RRT2LZXN"},"agent_actions":{"view_html":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU","download_json":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU.json","view_paper":"https://pith.science/paper/RRT2LZXN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08159&json=true","fetch_graph":"https://pith.science/api/pith-number/RRT2LZXNTVU6FNWRGNAE3EUBXU/graph.json","fetch_events":"https://pith.science/api/pith-number/RRT2LZXNTVU6FNWRGNAE3EUBXU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU/action/storage_attestation","attest_author":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU/action/author_attestation","sign_citation":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU/action/citation_signature","submit_replication":"https://pith.science/pith/RRT2LZXNTVU6FNWRGNAE3EUBXU/action/replication_record"}},"created_at":"2026-06-09T01:05:28.555974+00:00","updated_at":"2026-06-09T01:05:28.555974+00:00"}