On the girth and diameter of generalized Johnson graphs

Agong, Louis Anthony; Amarra, Carmen; Caughman, John S.; Herman, Ari J.; Terada, Taiyo S.

doi:10.1016/j.disc.2017.08.022

“…The eigenvalues of the adjacency matrix of J(n, k, r) are given by the so-called Eberlein polynomials; this can be derived in terms of association schemes [3,8] or by representation theoretic means [2,16]. The distance in J(n, k, r) was determined by Agong, Amarra, Caughman, Herman and Terada [1]. In general, the graph J(n, k, r) is not distance regular.…”

Section: Generalized Johnson Graphsmentioning

confidence: 99%

Nonassociativity of the Norton Algebras of Some Distance Regular Graphs

Huang¹

2020

Electron. J. Combin.

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A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of Levstein, Maldonado and Penazzi. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by Huang, Mickey, and Xu.

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“…Second, since n − 1 is odd, every adjacent vertices x and x ′ in Q The following isomorphism is well known, cf. [1]. Proposition 4.3.…”

Section: On the Structure Of Exact Distance Graphs Of Hypercubesmentioning

confidence: 94%

“…, n} : |A| = k} and edge set {AB : |A ∩ B| = i}. The family of generalized Johnson graphs includes Kneser graphs K(n, k) = J(n, k, 0) (which themselves include the odd graphs J (2k + 1, k, 0)) and the Johnson graphs J(n, k, k − 1)) [1].…”

Section: Exact Distance Graphs Of Hypercubesmentioning

confidence: 99%

Exact Distance Graphs of Product Graphs

Brešar

¹

,

Gastineau

²

,

Klavžar

³

et al. 2019

Graphs and Combinatorics

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Given a graph G, the exact distance-p graph G [♮p] has V (G) as its vertex set, and two vertices are adjacent whenever the distance between them in G equals p. We present formulas describing the structure of exact distance-p graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-2 graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-p graphs of hypercubes Qn are also studied. As these graphs contain generalized Johnson graphs as induced subgraphs, we use some known and find some new constructions of their colorings. These constructions are applied for colorings of the exact distance-p graphs of hypercubes with the focus on the chromatic number of Q [♮p] n for p ∈ {n − 2, n − 3, n − 4}.

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“…The eigenvalues of the adjacency matrix of J(n, k, r) are given by the so-called Eberlein polynomials; this can be derived in terms of association schemes [3,8] or by representation theoretic means [2,16]. The distance in J(n, k, r) was determined by Agong, Amarra, Caughman, Herman and Terada [1]. In general, the graph J(n, k, r) is not distance regular.…”

Section: Hamming Graphsmentioning

confidence: 99%

Nonassociativity of the Norton Algebras of some distance regular graphs

Huang¹

2020

Preprint

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A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of Levstein, Maldonado and Penazzi. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by Huang, Mickey, and Xu.

show abstract

On the girth and diameter of generalized Johnson graphs

Cited by 13 publications

References 7 publications

Nonassociativity of the Norton Algebras of Some Distance Regular Graphs

Nonassociativity of the Norton Algebras of Some Distance Regular Graphs

Exact Distance Graphs of Product Graphs

Nonassociativity of the Norton Algebras of some distance regular graphs

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