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We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8. 相似文献
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Bo Zhou 《Southeast Asian Bulletin of Mathematics》2000,24(3):499-504
The parameter l(G) for a primitive digraph G introduced by Lewin is the minimum positive integer k for which there are walks of both lengths k and k + 1 from some vertex u to some vertex v. We obtain upper bounds on l(G) if G is primitive ministrong, or G is just primitive and not necessarily ministrong, or G is primitive symmetric. We also discuss the numbers attainable as l(G).AMS Subject Classification (2000): 05C20, 15A48Partially supported by the National Natural Science Foundation of China (19771040) and the Guangdong Provincial Natural Science Foundation of China (990447). 相似文献
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David S. Lyubshin 《Discrete Mathematics》2009,309(13):4343-4348
In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form for some non-negative integer n, where Q8 is the quaternion group and is the abelian group of order 2n and exponent 2. 相似文献
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Jian-qin Zhou 《应用数学学报(英文版)》2008,24(2):185-194
Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguil5 et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2,..., over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x,y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, ~ ~ ~ ) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ y - x ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches. 相似文献