There are only finitely many distance-regular graphs of fixed valency greater than two

Bang, Sejeong; Dubickas, Artūras; Koolen, Jack H.; Moulton, Vincent

doi:10.1016/j.aim.2014.09.025

Cited by 30 publications

(39 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In turn, these give rise to infinite families of cubic 5-walk-regular graphs and 7-walk-regular graphs with valency 4. The validity of the Bannai-Ito conjecture [1] (in particular the fact that there are finitely many distance-regular graphs with valency four [6]) for example implies that there are infinitely many 7-walk-regular graphs that are not distance-regular.…”

Section: Construction Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Geometric aspects of 2-walk-regular graphs

Cámara

Dam

Koolen

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show

show abstract

Section: Construction Methodsmentioning

confidence: 99%

“…If c 2 = 1, then n ≥ 1 + k + k 2 = 13, a contradiction. If c 2 = 2, then n ≥ 1 + k + k 2 = 9, with equality if and only if Γ is strongly regular with parameters (9,4,1,2). This implies that Γ is the lattice graph L 2 (3), which however has no eigenvalue with multiplicity 3.…”

Section: Small Multiplicitymentioning

confidence: 99%

Geometric aspects of 2-walk-regular graphs

Cámara

Dam

Koolen

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…As applications, some upper bounds are derived for the diameter d (= dimension −1) of the algebra. In the case of an integral table algebra, d is bounded as a function of d − g (where g is the number of repeated columns) and the valency k. This generalizes the results of Bannai and Ito [4][5][6] on the adjacency algebra of a distance-regular graph, as well as a special case of the solution [2] of the Bannai-Ito conjecture for distance-regular graphs. Several definitions and remarks must be given in order to state our results precisely.…”

Section: Introductionmentioning

confidence: 62%

“…The adjacency algebra (Bose-Mesner algebra) of a distance-regular graph satisfies the hypotheses of Theorem 1.6 (if there are at least two repeated columns). So the theorem generalizes part of Bannai and Ito's results in [6], which are of course superseded by [2]. Theorem 1.6.…”

Section: Introductionmentioning

confidence: 74%

On the Calculation of Multiplicities forP-PolynomialC-Algebras

Blau

Hein²

2013

Communications in Algebra

View full text Add to dashboard Cite

An alternative rational function, with polynomial components of smaller degree, is constructed to compute multiplicities for a P-polynomial C-algebra whose generating tri-diagonal matrix has a set of repeated column entries. As a consequence, some upper bounds are derived for the diameter of the algebra. The bound in the case of an integral table algebra generalizes a well known result of Bannai and Ito for distance-regular graphs.

show abstract

“…Also applications of the symmetry properties of parallel lines and the main areas where such cutting opportunities should be implemented central symmetry circles (Banga, S. et al, 2014;Burdette, A. C., 1971). On the diagram of Figure 1 shows the placement of the circumscribed circle of hexagons in a spherical triangle (compatible segment of the sphere) with interior angles 36, 90 and 60 °.…”

Section: The Main Partmentioning

confidence: 99%