Journal ArticleDOI
Edge-symmetric distance-regular coverings of cliques: The affine case
TLDR
In this article, the authors proposed a classification for the graphs based on the description of twice transitive permutation groups for affine bipartite clique coverings of edge-symmetric distance-regular covering of cliques.Abstract:
Let Γ be an edge-symmetric distance-regular covering of a clique. Then the group G = Aut(Γ) acts twice transitively on the set Σ of antipodal classes. We propose a classification for the graphs based on the description of twice transitive permutation groups. This program is realized for a1 = c2. In this article we classify graphs in the case when the action of G on Σ is affine.read more
Citations
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Arc-transitive antipodal distance-regular covers of complete graphs related to SU3(q)
TL;DR: A new infinite family of distance-regular antipodal r -covers of a complete graph on q 3 + 1 vertices is found, where q is odd and r is any divisor of q + 1 such that (q + 1 ) ∕ r is odd.
Journal ArticleDOI
The Existence of Perfect Codes in Doob Graphs
TL;DR: In this article, it was shown that 1-perfect codes in the Doob graph exist if and only if $6 {m}+3 {n}+1$ is a power of 2; that is, if the size of a 1-ball divides the number of vertices.
Journal ArticleDOI
Edge-Symmetric Distance-Regular Coverings of Complete Graphs: the Almost Simple Case
TL;DR: In this paper, the classification of edge-symmetric distance-regular coverings of complete graphs with r /∉ {2, k, (k − 1)/μ} for the case of the almost simple action of an automorphism group of a graph on a set of its antipodal classes is complete.
Journal ArticleDOI
The existence of perfect codes in Doob graphs
TL;DR: It is shown that 1-perfect codes in the Doob graph exist if and only if the size of a 1-ball divides the number of vertices is a power of 2.
References
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Book
Distance-Regular Graphs
TL;DR: In this paper, a connected simple graph with vertex set X of diameter d is considered, and the authors define Ri X2 by (x, y) Ri whenever x and y have graph distance.
Journal ArticleDOI
Finite permutation groups and finite simple groups
TL;DR: In this paper, the authors consider the theory of finite permutation groups with the assumption that the finite simple groups are known, and examine questions such as: which problems are solved or solvable under this assumption, and what important problems remain?