There Does Not Exist a Distance-Regular Graph with Intersection Array { 80 , 54 , 12 ; 1 , 6 , 60 }

Jack H. Koolen; Quaid Iqbal; Jongyook Park; Masood Ur Rehman

doi:10.1007/s00373-019-02095-2

There Does Not Exist a Distance-Regular Graph with Intersection Array { 80 , 54 , 12 ; 1 , 6 , 60 }

Jack H. Koolen, Quaid Iqbal, Jongyook Park, Masood Ur Rehman

University of Science and Technology of China

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

Abstract

In this paper, we will show that there does not exist a distance-regular graph Γ with intersection array { 80 , 54 , 12 ; 1 , 6 , 60 }. We first show that a local graph Δ of Γ does not contain a coclique with 5 vertices, and then we prove that the graph Γ is geometric by showing that Δ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph Γ , and we could obtain that there is no such a distance-regular graph.

Original language	English
Pages (from-to)	1597-1608
Number of pages	12
Journal	Graphs and Combinatorics
Volume	35
Issue number	6
DOIs	https://doi.org/10.1007/s00373-019-02095-2
State	Published - 1 Nov 2019

Keywords

Delsarte cliques
Distance-regular graphs
Geometric distance-regular graphs
The claw-bound

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title = "There Does Not Exist a Distance-Regular Graph with Intersection Array { 80 , 54 , 12 ; 1 , 6 , 60 }",

abstract = "In this paper, we will show that there does not exist a distance-regular graph Γ with intersection array { 80 , 54 , 12 ; 1 , 6 , 60 }. We first show that a local graph Δ of Γ does not contain a coclique with 5 vertices, and then we prove that the graph Γ is geometric by showing that Δ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph Γ , and we could obtain that there is no such a distance-regular graph.",

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AU - Iqbal, Quaid

AU - Park, Jongyook

AU - Ur Rehman, Masood

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AB - In this paper, we will show that there does not exist a distance-regular graph Γ with intersection array { 80 , 54 , 12 ; 1 , 6 , 60 }. We first show that a local graph Δ of Γ does not contain a coclique with 5 vertices, and then we prove that the graph Γ is geometric by showing that Δ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph Γ , and we could obtain that there is no such a distance-regular graph.

KW - Delsarte cliques

KW - Distance-regular graphs

KW - Geometric distance-regular graphs

KW - The claw-bound

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JF - Graphs and Combinatorics

IS - 6

ER -

There Does Not Exist a Distance-Regular Graph with Intersection Array { 80 , 54 , 12 ; 1 , 6 , 60 }

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Keywords

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