# The Laplacian spectral excess theorem for distance-regular graphs

E.R. van Dam, M.A. Fiol

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.
Original language English 245-250 Linear Algebra and its Applications 458 24 Jun 2014 https://doi.org/10.1016/j.laa.2014.06.001 Published - 1 Oct 2014

### Fingerprint

Distance-regular Graph
Excess
Theorem
Laplacian Spectrum
Regular Graph
Equality
Regularity
If and only if
Vertex of a graph

### Keywords

• Distance-regular graphs
• spectral excess theorem
• Laplacian spectrum
• orthogonal polynomials

### Cite this

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title = "The Laplacian spectral excess theorem for distance-regular graphs",
abstract = "The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.",
keywords = "Distance-regular graphs, spectral excess theorem, Laplacian spectrum, orthogonal polynomials",
author = "{van Dam}, E.R. and M.A. Fiol",
year = "2014",
month = "10",
day = "1",
doi = "10.1016/j.laa.2014.06.001",
language = "English",
volume = "458",
pages = "245--250",
journal = "Linear Algebra and its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",

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In: Linear Algebra and its Applications, Vol. 458, 01.10.2014, p. 245-250.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - The Laplacian spectral excess theorem for distance-regular graphs

AU - van Dam, E.R.

AU - Fiol, M.A.

PY - 2014/10/1

Y1 - 2014/10/1

N2 - The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.

AB - The spectral excess theorem states that, in a regular graph Γ, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of Γ), and Γ is distance-regular if and only if equality holds. In this note we prove the corresponding result by using the Laplacian spectrum without requiring regularity of Γ.

KW - Distance-regular graphs

KW - spectral excess theorem

KW - Laplacian spectrum

KW - orthogonal polynomials

U2 - 10.1016/j.laa.2014.06.001

DO - 10.1016/j.laa.2014.06.001

M3 - Article

VL - 458

SP - 245

EP - 250

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -