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Jurišić, Aleksandar; Koolen, Jack H.

doi:10.1023/a:1025133213542

Cited by 17 publications

(6 citation statements)

References 16 publications

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“…Remark 11. In the proof of Theorem 8, we saw that the µ-graph has the eigenvalue λ := ντ −1 when equality holds in (11). By the equation on the left in ( 14) it follows that λ = (σq − rp)τ −1 .…”

Section: A New Feasibility Conditionmentioning

confidence: 95%

“…Verify pq − r(p + q) + q 3 > 0 by considering each case of Lemma 5. Using this inequality, solve ( 16) for r and simplify the result to obtain (11).…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

“…For the second assertion, the equality in (11) holds if and only if the equality in (15) holds if and only if there exists ν ∈ R such that λ i = ντ −1 for all i (1 i τ ). Thus, if the equality holds in (11), then we find that the µ-graph has three distinct eigenvalues: p, −q, and ντ −1 . Therefore, the µ-graph is strongly regular.…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

“…For example, the µ-graph of AT4(2, 2, 2) is K 2,2 , which is a strongly regular graph. However, equality in (11) does not hold since r = 2 = 5/2 = (p + q 3 )(p + q) −1 .…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

“…In Section 3, we gave a new feasibility condition (11) for the AT4(p, q, r) family. In this section, we give some comments on the graphs AT4(p, q, r) when equality holds in (11), that is, r = (p + q 3 )(p + q) −1 . Let Γ denote an antipodal tight graph AT4(p, q, r) with r = (p + q 3 )(p + q) −1 .…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

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A New Feasibility Condition for the AT4 Family

Xia¹,

Lee²,

Koolen³

2023

Electron. J. Combin.

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Let $\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$. Then $\Gamma$ is tight in the sense of Jurišić, Koolen, and Terwilliger (J. Algebraic Combin, 2000) whenever $\Gamma$ is locally strongly regular with nontrivial eigenvalues $p:=\theta_2$ and $-q:=\theta_3$. Assume that $\Gamma$ is tight. Then the intersection numbers of $\Gamma$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $\Gamma$. We denote $\Gamma$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph.Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.

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Section: A New Feasibility Conditionmentioning

confidence: 95%

“…Verify pq − r(p + q) + q 3 > 0 by considering each case of Lemma 5. Using this inequality, solve ( 16) for r and simplify the result to obtain (11).…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

Section: A New Feasibility Conditionmentioning

confidence: 99%

“…For example, the µ-graph of AT4(2, 2, 2) is K 2,2 , which is a strongly regular graph. However, equality in (11) does not hold since r = 2 = 5/2 = (p + q 3 )(p + q) −1 .…”

Section: A New Feasibility Conditionmentioning

confidence: 99%

Section: A New Feasibility Conditionmentioning

confidence: 99%

See 3 more Smart Citations

A New Feasibility Condition for the AT4 Family

Xia¹,

Lee²,

Koolen³

2023

Electron. J. Combin.

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Proper Lucky Labeling of Jelly Fish Graph, Cocktail Party Graph and Crown Graph

Kumar

Meenakshi

2022

Proceedings of 2nd International Conference on Mathematical Modeling and Computational Science

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The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Suzuki

2005

J Algebr Comb

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Abstract. Let be a distance-regular graph of diameter D. Let X denote the vertex set of and let Y be a nonempty subset of X . We define an algebra T = T (Y ). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then T is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible T -modules. We define endpoints and thin condition on irreducible T -modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y , it is thin if and only if Y is a completely regular code of . By considering a suitable subset Y , every irreducible T (x)-module of endpoint i can be regarded as an irreducible T (Y )-module of endpoint 0.

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Cited by 17 publications

References 16 publications

A New Feasibility Condition for the AT4 Family

A New Feasibility Condition for the AT4 Family

Proper Lucky Labeling of Jelly Fish Graph, Cocktail Party Graph and Crown Graph

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

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