Pith Number

pith:QNALD3G3

pith:2026:QNALD3G3RDFIBAVFKMTRZ7FREV

not attested not anchored not stored refs pending

Local Tur\'an inequalities for walks and the spectral radius

Feng Liu, Qi Wu, Shuang Sun, Yan Wang

For any graph G and integer r at least 1, the spectral radius to the power r is at most the sum over vertices of the number of r-walks from each vertex times one minus the reciprocal of its largest clique size.

arxiv:2605.02191 v2 · 2026-05-04 · math.CO

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\usepackage{pith}
\pithnumber{QNALD3G3RDFIBAVFKMTRZ7FREV}

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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

For every finite simple graph G and every integer r ≥ 1, λ₁(G)^r ≤ ∑_{v∈V(G)} w_r(v) (c_G(v)−1)/c_G(v). This confirms a conjecture of Kannan, Kumar, and Pragada.

C2weakest assumption

The proof invokes a weighted local spectral Turán theorem whose precise statement and proof are not visible in the abstract; if that auxiliary result fails to hold in the required weighted form, the main inequality does not follow.

C3one line summary

The inequality λ₁(G)^r ≤ ∑ w_r(v) ⋅ (c_G(v)−1)/c_G(v) holds for every finite simple graph G and r ≥ 1, confirming the Kannan-Kumar-Pragada conjecture.

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

On spectral Tur\'an theorems: confirming a conjecture of Guiduli and two problems of Nikiforov

Receipt and verification

First computed	2026-06-09T01:05:18.643424Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8340b1ecdb88ca8082a553271cfcb12557fe585bda7fce80f13e0da8c59dba18

Aliases

arxiv: 2605.02191 · arxiv_version: 2605.02191v2 · doi: 10.48550/arxiv.2605.02191 · pith_short_12: QNALD3G3RDFI · pith_short_16: QNALD3G3RDFIBAVF · pith_short_8: QNALD3G3

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/QNALD3G3RDFIBAVFKMTRZ7FREV \
  | 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: 8340b1ecdb88ca8082a553271cfcb12557fe585bda7fce80f13e0da8c59dba18

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "123667773630c42d7f26ff9ded9ddf9238d1582eff3945d8afdc5e5e807140df",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-04T03:42:18Z",
    "title_canon_sha256": "4ff1f627aea96aa9ca5fdc6316e6c0f3361d07c5025cd9dd95543ef448400176"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.02191",
    "kind": "arxiv",
    "version": 2
  }
}