{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FM6LK4HTU3KKGK7XMTJ6EA2UEW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a517cf75a63ce02e47dfe6306b3234114061d5842a893ad0f067719d04ba2c45","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-09-30T02:55:00Z","title_canon_sha256":"22f271d819cc1a4c9db700644bb955803e7a78f5aa32b04d5fc18db2ab9b6fce"},"schema_version":"1.0","source":{"id":"1509.09000","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.09000","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"arxiv_version","alias_value":"1509.09000v1","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.09000","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"pith_short_12","alias_value":"FM6LK4HTU3KK","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FM6LK4HTU3KKGK7X","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FM6LK4HT","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:83ac8b4589e7254cde873eacf4691cd5454e4bb54e2b1cf1334a9f0f3050f8cd","target":"graph","created_at":"2026-05-18T01:31:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and removing infinitely many vertices and edges. Using this result, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from $\\mathbb{Z}^2$ by randomly adding vertices.","authors_text":"Akito Suzuki, Itaru Sasaki","cross_cats":["math.MP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-09-30T02:55:00Z","title":"Essential spectrum of the discrete Laplacian on a perturbed periodic graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09000","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:74ba9399d2dd0a56c2210f5f42fee22ab8c35d2d2d1a28a723a6d496cea17609","target":"record","created_at":"2026-05-18T01:31:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a517cf75a63ce02e47dfe6306b3234114061d5842a893ad0f067719d04ba2c45","cross_cats_sorted":["math.MP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-09-30T02:55:00Z","title_canon_sha256":"22f271d819cc1a4c9db700644bb955803e7a78f5aa32b04d5fc18db2ab9b6fce"},"schema_version":"1.0","source":{"id":"1509.09000","kind":"arxiv","version":1}},"canonical_sha256":"2b3cb570f3a6d4a32bf764d3e203542590615108e711d40728195423135a8f15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b3cb570f3a6d4a32bf764d3e203542590615108e711d40728195423135a8f15","first_computed_at":"2026-05-18T01:31:29.331097Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:29.331097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g9sO6iVj+P0c6W4u5aRJujqaCd8K+wOjkHhvCp+rImWgapDn1/NllOryvBgZoTfcL7EeLkGs1FQxVOfcw4eXBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:29.331513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.09000","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74ba9399d2dd0a56c2210f5f42fee22ab8c35d2d2d1a28a723a6d496cea17609","sha256:83ac8b4589e7254cde873eacf4691cd5454e4bb54e2b1cf1334a9f0f3050f8cd"],"state_sha256":"5256419600c28a4490bd2a9238dfcbfcd23a37a363c00d1c36872c23cae66eb7"}