Pith Number

pith:JCQKMZ2F

pith:2012:JCQKMZ2FGRUNLX4ZNURFOQPBL6

not attested not anchored not stored refs pending

Sharp upper and lower bounds on the number of spanning trees in Cartesian product of graphs

Jernej Azarija

arxiv:1210.6340 v1 · 2012-10-23 · math.CO

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{JCQKMZ2FGRUNLX4ZNURFOQPBL6}

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

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	2026-05-18T03:42:32.656488Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

48a0a667453468d5df996d225741e15fa1af979d997987e9f4f3f129a7308d73

Aliases

arxiv: 1210.6340 · arxiv_version: 1210.6340v1 · doi: 10.48550/arxiv.1210.6340 · pith_short_12: JCQKMZ2FGRUN · pith_short_16: JCQKMZ2FGRUNLX4Z · pith_short_8: JCQKMZ2F

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/JCQKMZ2FGRUNLX4ZNURFOQPBL6 \
  | 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: 48a0a667453468d5df996d225741e15fa1af979d997987e9f4f3f129a7308d73

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "053a78876bd4caad49515f31ccc308930df76abc634b8bc4bc0321133457f402",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2012-10-23T19:47:15Z",
    "title_canon_sha256": "28a98268a70f9141100389a868ae30afaab1b47d97460e8582346ca9fde45f83"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1210.6340",
    "kind": "arxiv",
    "version": 1
  }
}