Pith Number

pith:XBUSLIF3

pith:2026:XBUSLIF35WBEL6MXOUCNWZOQOS

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Extremal chromatic bounds for distance Laplacian eigenvalues

Bilal Ahmad Rather

Color class sizes from an optimal coloring bound the first several distance Laplacian eigenvalues of a connected graph.

arxiv:2604.10785 v2 · 2026-04-12 · math.CO · cs.DM

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Claims

C1strongest claim

If (ℓ1,…,ℓχ) are the color-class sizes in an optimal χ-coloring with ℓ1≥⋯≥ℓχ, then the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G)≥n+ℓ1 for 1≤i≤ℓ1−1.

C2weakest assumption

The graph is connected and the given partition is an optimal proper χ-coloring whose class sizes are ordered decreasingly; the proof relies on this ordering to apply the majorization step to the distance matrix.

C3one line summary

Proves that for a χ-chromatic graph the first ℓ1−1 distance Laplacian eigenvalues satisfy ∂^L_i(G) ≥ n + ℓ1 where ℓ1 is the largest color-class size, refining distribution and extremal results.

References

15 extracted · 15 resolved · 0 Pith anchors

[1] M. Aouchiche and P. Hansen, Two Laplacians for the distance matrix of a graph,Linear Algebra Appl.439(2013) 21–33 2013

[2] M. Aouchiche and P. Hansen, Some properties of the distance Laplacian eigenvalues of a graph,Czechoslovak Math. J.64(139) (2014) 751–761 2014

[3] M. Aouchiche and P. Hansen, Distance Laplacian eigenvalues and chromatic number in graphs,Filomat31(9) (2017) 2545–2555 2017

[4] A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Springer, New York, 2012 2012

[5] D. Cvetkovi´ c, M. Doob, and H. Sachs,Spectra of Graphs – Theory and Applications, 3rd ed., Johann Ambrosius Barth Verlag, Heidelberg–Leipzig, 1995 1995

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:37.840974Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b86925a0bbed8245f9977504db65d074bfd3a9e71817369dc12c156c52cd715d

Aliases

arxiv: 2604.10785 · arxiv_version: 2604.10785v2 · doi: 10.48550/arxiv.2604.10785 · pith_short_12: XBUSLIF35WBE · pith_short_16: XBUSLIF35WBEL6MX · pith_short_8: XBUSLIF3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/XBUSLIF35WBEL6MXOUCNWZOQOS \
  | 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: b86925a0bbed8245f9977504db65d074bfd3a9e71817369dc12c156c52cd715d

Canonical record JSON

{
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    "abstract_canon_sha256": "84ab21c81cdd029d0294f1f0d687836c233ea83ed2f9d6a40fd3cc983d50bb8f",
    "cross_cats_sorted": [
      "cs.DM"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-04-12T19:22:19Z",
    "title_canon_sha256": "c54740941a9a3925309c9a7a3db2654a89e748454c4bc60479663edceb0415a5"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 2
  }
}