Pith Number

pith:HFQJM7JM

pith:2011:HFQJM7JM72MIMUM2ZOTCACDJSA

not attested not anchored not stored refs pending

Expanders Are Universal for the Class of All Spanning Trees

Daniel Johannsen, Michael Krivelevich, Wojciech Samotij

arxiv:1108.4647 v1 · 2011-08-23 · math.CO

Open paper page JSON Open Graph Bundle Merged state What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{HFQJM7JM72MIMUM2ZOTCACDJSA}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-05-18T04:14:54.029714Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3960967d2cfe9886519acba62008699021597696a2b5af5af4607743fa48402b

Aliases

arxiv: 1108.4647 · arxiv_version: 1108.4647v1 · doi: 10.48550/arxiv.1108.4647 · pith_short_12: HFQJM7JM72MI · pith_short_16: HFQJM7JM72MIMUM2 · pith_short_8: HFQJM7JM

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/HFQJM7JM72MIMUM2ZOTCACDJSA \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3960967d2cfe9886519acba62008699021597696a2b5af5af4607743fa48402b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b560127dc9e8622c472917017614acdfdca24a46eed960069d75bf677837be23",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2011-08-23T16:14:56Z",
    "title_canon_sha256": "c9b9e255ecdefe3983177f62976e823288282f809dee5bf9a046cabdf0b1115f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1108.4647",
    "kind": "arxiv",
    "version": 1
  }
}