An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

Jack H. Koolen; Jongyook Park; Hyonju Yu

doi:10.1016/j.laa.2010.12.032

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

Jack H. Koolen
, Jongyook Park
, Hyonju Yu

Pohang University of Science and Technology

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)≤-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

Original language	English
Pages (from-to)	2404-2412
Number of pages	9
Journal	Linear Algebra and Its Applications
Volume	434
Issue number	12
DOIs	https://doi.org/10.1016/j.laa.2010.12.032
State	Published - 15 Jun 2011

Keywords

Bounds on eigenvalues
Distance-regular graph
Shill distance-regular graphs
Tight distance-regular graph

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10.1016/j.laa.2010.12.032

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title = "An inequality involving the second largest and smallest eigenvalue of a distance-regular graph",

abstract = "For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)≤-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.",

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AU - Koolen, Jack H.

AU - Park, Jongyook

AU - Yu, Hyonju

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N2 - For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)≤-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

AB - For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)≤-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

KW - Bounds on eigenvalues

KW - Distance-regular graph

KW - Shill distance-regular graphs

KW - Tight distance-regular graph

UR - https://www.scopus.com/pages/publications/79952628984

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DO - 10.1016/j.laa.2010.12.032

M3 - Article

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 12

ER -

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

Abstract

Keywords

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