Distance-regular graphs with Γ(x)≃3∗Ka+1

Yamazaki, Norio

doi:10.1016/0195-6698(95)90006-3

Cited by 10 publications

(15 citation statements)

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“…Moreover, using the fact that if is triangle-free then l (1,1,k−2) ≤ 3 holds [11], we can also extend our proof of Theorem 1.2 to show that there are finitely many triangle-free distance-regular graphs that have degrees 8 or 9. An extension of the proof of Corollary 1.2 can then be used to show that there are finitely many distance-regular graphs with degrees 8 and 9 (indeed, the only case that needs to be considered is k = 8 and a 1 = 1, since this is not covered by the results of Yamazaki given in [17]). …”

Section: Resultsmentioning

confidence: 98%

“…In addition, the Bannai-Ito conjecture has been shown to hold for several special classes of distance-regular graphs: for example, Bannai and Ito showed that it holds for the class of bipartite distance-regular graphs, Mohar and Shawe-Taylor [15] and independently Brouwer [8,Theorem 4.2.16] showed that it holds for the class of distanceregular line graphs that are not polygons, and Hiraki et al [12] and Yamazaki [17] showed that it holds for the class of distance-regular graphs with k = 3(a 1 + 1), a 1 ≥ 1.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

On a Conjecture of Bannai and Ito: There are Finitely Many Distance-regular Graphs with Degree 5, 6 or 7

Koolen

Moulton

2002

European Journal of Combinatorics

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Section: Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

On a Conjecture of Bannai and Ito: There are Finitely Many Distance-regular Graphs with Degree 5, 6 or 7

Koolen

Moulton

2002

European Journal of Combinatorics

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“…Yamazaki [21] considered distance-regular graphs which are locally a disjoint union of three cliques of size a 1 + 1, and for a 1 1 these graphs are geometric distance-regular graphs with smallest eigenvalue −3.…”

Section: Introductionmentioning

confidence: 99%

Geometric distance-regular graphs without 4-claws

Bang

2013

Linear Algebra and its Applications

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A non-complete distance-regular graph is called geometric if there exists a set C of Delsarte cliques such that each edge of lies in a unique clique in C. In this paper we determine the non-complete distance-regular graphs satisfying max 3, 8 3 (a 1 + 1) < k < 4a 1 + 10 − 6c 2 . To prove this result, we first show by considering nonexistence of 4-claws that any non-complete distance-regular graph satisfying max 3, 8 3 (a 1 + 1) < k < 4a 1 + 10 − 6c 2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, two feasible intersection arrays in the list of [7, Chapter 14] are ruled out.

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“…Hence θ 3 = −5 and thus 2 ≤ ψ 1 ≤ 4 and ψ 1 ≥ Remark 4.6 It seems to be difficult to classify distance-regular graphs with θ D = −3 satisfying k = 3(a 1 + 1) and c 2 = 1 (see [4,38]). It follows by [4], [7] and [38] that any distance-regular graph in Theorem 4.5 (9) (i.e., a distance-regular graph with θ D = −3 satisfying k = 3(a 1 + 1), c 2 = 1 and a 1 , …”

Section: Since Parametersmentioning

confidence: 99%

“…Hiraki, Nomura and Suzuki [24] and Yamazaki [38] considered distance-regular graphs that are locally a disjoint union of three cliques of size a 1 + 1 (i.e., 4-claw-free), and these graphs for a 1 ≥ 1 turn out to be geometric with smallest eigenvalue −3 (see Remark 4.6). The problem of classification of such graphs seems to be very hard.…”

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confidence: 99%