NDLI: Tight Distance-Regular Graphs

Content Provider	Springer Nature Link
Author	Jurišić, Aleksandar Koolen, Jack Terwilliger, Paul
Copyright Year	2000
Abstract	We consider a distance-regular graph Γ with diameter d ≥ 3 and eigenvalues k = θ$_{0}$ > θ$_{1}$ > ... > θ$_{ d }$. We show the intersection numbers a $_{1}$, b $_{1}$ satisfy $$\left( {\theta _1 + \frac{k}{{a_1 + 1}}} \right)\left( {\theta _d + \frac{k}{{a_1 + 1}}} \right) \geqslant - \frac{{ka_1 b_1 }}{{(a_1 + 1)^2 }}.$$ We say Γ is tight whenever Γ is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show Γ is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show Γ is tight if and only if a $_{1}$ ≠ 0, a $_{d}$ = 0, and Γ is 1-homogeneous in the sense of Nomura. We show Γ is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues −1 − b $_{1}$(1 + θ$_{1}$)$^{−1}$ and −1 − b $_{1}$(1 + θ$_{ d }$)$^{−1}$. Three infinite families and nine sporadic examples of tight distance-regular graphs are given.
Starting Page	163
Ending Page	197
Page Count	35
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	12
Issue Number	2
e-ISSN	15729192
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2000-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computer Science Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Combinatorics Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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