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Graph $G$ is called a \\emph{$k$-DM} if it has a $k$-DM labeling(L). Long Brush is a graph $G$ with $V(G)$ = $\\{u_1, u_2, . . . , u_n,$ $v_1, v_2, . . . , v_{m}\\}$, a path $P_n$ = $u_1$ $u_2$ . . . $u_n$ and $E(G)$ = $E(P_n)$ $\\cup$ $\\{u_1v_i:$ $i$ = 1 to $m\\}$ $\\cup$ $E(<v_1, v_2, . . . , v_{m}>)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2211.09666","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2022-11-13T17:49:19Z","cross_cats_sorted":[],"title_canon_sha256":"9c7b3d63f1461ff86fce1d3972b72368aab154eeb8ff44b4a8287ae3d81572be","abstract_canon_sha256":"c5998c38325fe559d481dc931d7f922cf116fcbb261663b776ea1e42e7361cf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:17:30.394757Z","signature_b64":"sgPbDasO1WFAUFag1Flj2WmpKkC3DYfvgW/kaqjyCZvenx7ER2C530UC9J9cbDIlxV+hEdBmPLzD20tkwtk1AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8803d5e90388f46586811143a0fb213992c3251f0885ae48e656e01ac0c08099","last_reissued_at":"2026-07-05T05:17:30.394438Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:17:30.394438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$k$-Distance Magic Labeling and Long Brush Graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"V. Vilfred Kamalappan","submitted_at":"2022-11-13T17:49:19Z","abstract_excerpt":"We define a labeling $f:$ $V(G)$ $\\rightarrow$ $\\{1, 2, \\ldots, n\\}$ on a graph $G$ of order $n \\geq 3$ as a \\emph{$k$-distance magic} ($k$-DM) if $\\sum_{w\\in \\partial N_k(u)}{ f(w)}$ is a constant and independent of $u\\in V(G)$ where $\\partial N_k(u)$ = $\\{v\\in V(G): d(u, v) = k\\}$, $k\\in\\mathbb{N}$. Graph $G$ is called a \\emph{$k$-DM} if it has a $k$-DM labeling(L). Long Brush is a graph $G$ with $V(G)$ = $\\{u_1, u_2, . . . , u_n,$ $v_1, v_2, . . . , v_{m}\\}$, a path $P_n$ = $u_1$ $u_2$ . . . $u_n$ and $E(G)$ = $E(P_n)$ $\\cup$ $\\{u_1v_i:$ $i$ = 1 to $m\\}$ $\\cup$ $E(<v_1, v_2, . . . , v_{m}>)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.09666","kind":"arxiv","version":2},"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/2211.09666/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":"2211.09666","created_at":"2026-07-05T05:17:30.394485+00:00"},{"alias_kind":"arxiv_version","alias_value":"2211.09666v2","created_at":"2026-07-05T05:17:30.394485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.09666","created_at":"2026-07-05T05:17:30.394485+00:00"},{"alias_kind":"pith_short_12","alias_value":"RAB5L2IDRD2G","created_at":"2026-07-05T05:17:30.394485+00:00"},{"alias_kind":"pith_short_16","alias_value":"RAB5L2IDRD2GLBUB","created_at":"2026-07-05T05:17:30.394485+00:00"},{"alias_kind":"pith_short_8","alias_value":"RAB5L2ID","created_at":"2026-07-05T05:17:30.394485+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/RAB5L2IDRD2GLBUBCFB2B6ZBHG","json":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG.json","graph_json":"https://pith.science/api/pith-number/RAB5L2IDRD2GLBUBCFB2B6ZBHG/graph.json","events_json":"https://pith.science/api/pith-number/RAB5L2IDRD2GLBUBCFB2B6ZBHG/events.json","paper":"https://pith.science/paper/RAB5L2ID"},"agent_actions":{"view_html":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG","download_json":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG.json","view_paper":"https://pith.science/paper/RAB5L2ID","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2211.09666&json=true","fetch_graph":"https://pith.science/api/pith-number/RAB5L2IDRD2GLBUBCFB2B6ZBHG/graph.json","fetch_events":"https://pith.science/api/pith-number/RAB5L2IDRD2GLBUBCFB2B6ZBHG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG/action/storage_attestation","attest_author":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG/action/author_attestation","sign_citation":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG/action/citation_signature","submit_replication":"https://pith.science/pith/RAB5L2IDRD2GLBUBCFB2B6ZBHG/action/replication_record"}},"created_at":"2026-07-05T05:17:30.394485+00:00","updated_at":"2026-07-05T05:17:30.394485+00:00"}