Pith Number

pith:6C4DF7SM

pith:2025:6C4DF7SMBM32PPHLWQN2JH37X7

not attested not anchored not stored refs pending

Automorphism groups and Distinguishing Colorings of Central and Middle Graphs

Alexa Gopaulsingh, Amitayu Banerjee, Zal\'an Moln\'ar

If a graph G has at least four vertices, then Aut(G) is isomorphic to Aut(C(G)) and Aut(M(G)).

arxiv:2507.16301 v3 · 2025-07-22 · math.CO · math.GR

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\usepackage{pith}
\pithnumber{6C4DF7SMBM32PPHLWQN2JH37X7}

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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.

Claims

C1strongest claim

If the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups).

C2weakest assumption

The constructions of C(G) and M(G) from the subdivision graph S(G) preserve the full automorphism group of G when |V(G)| >= 4; this is the load-bearing step that allows the isomorphism to be claimed and the distinguishing bounds to follow.

C3one line summary

For connected graphs G with |V(G)| >= 4, Aut(G) ≅ Aut(C(G)) ≅ Aut(M(G)) as abstract groups, which yields new upper bounds on the distinguishing number and index of the central and middle graphs.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-06-19T16:09:50.351496Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f0b832fe4c0b37a7bcebb41ba49f7fbfccff0d8db2dcbf4ec6d75c2082b71b6f

Aliases

arxiv: 2507.16301 · arxiv_version: 2507.16301v3 · doi: 10.48550/arxiv.2507.16301 · pith_short_12: 6C4DF7SMBM32 · pith_short_16: 6C4DF7SMBM32PPHL · pith_short_8: 6C4DF7SM

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/6C4DF7SMBM32PPHLWQN2JH37X7 \
  | 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: f0b832fe4c0b37a7bcebb41ba49f7fbfccff0d8db2dcbf4ec6d75c2082b71b6f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "36f2f34241c1c5dfca22ef9d9e1a3da3d0806ae8f0daa0daea0d18afc9164bfe",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2025-07-22T07:39:10Z",
    "title_canon_sha256": "18503ca3409ccc39404238e958c3a7223a99712a9970408280e7ef88b9159252"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.16301",
    "kind": "arxiv",
    "version": 3
  }
}