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We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph $G$, $\\reg(R/I(G))\\leq \\min\\{a'(G),n(G)\\}$ and for a shellable graph $G$, $\\reg(R/I(G))\\leq n(G)$. Moreover it is shown that for a graph $G$, where $G^c$ is a $d$-tree, we have $\\pd(R/I(G))=\\max_{v\\in V(G)} \\{\\deg_G(v)\\}$."},"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":"1007.4056","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-07-23T07:27:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"13fb17a17ab139b2625a753dbc9e52bf2c8ecab04052add18abdd695e85d82ad","abstract_canon_sha256":"27b75fda2b5811ca4fb43d7a6e0ebfcb6127cdc6c8ced32e5ee65639199d0a11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:31.821338Z","signature_b64":"9CUgvvT6yl/DUsz7jKVs6E4J2zwVWfpovekHE4xJZSiTtGJ8UrRgM/H+H2zU3nounFLZJ/5vL0LCv5rMVEMyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2805dd4bdf784a292656649dc7efbb84b2a5fd9ba8324709b5695ef4e1e2fdf5","last_reissued_at":"2026-05-18T01:23:31.820824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:31.820824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds for the regularity of edge ideal of vertex decomposable and shellable graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Dariush Kiani, Somayeh Moradi","submitted_at":"2010-07-23T07:27:49Z","abstract_excerpt":"In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. 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