Cayley Graphs
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Gaojun Luo ◽  
Xiwang Cao ◽  
Guangkui Xu ◽  
Yingjie Cheng

Nikolai Nøjgaard ◽  
Walter Fontana ◽  
Marc Hellmuth ◽  
Daniel Merkle

2021 ◽  
Vol 28 (2) ◽  
Jia Huang

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

2021 ◽  
pp. 103383
Peter Bradshaw ◽  
Seyyed Aliasghar Hosseini ◽  
Jérémie Turcotte

Yan-Ting Xie ◽  
Yong-De Feng ◽  
Shoujun Xu

A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs —Cayley graphs generated by transpositions and show that a Cayley graph Γ generated by transpositions is a partial cube if and only if Γ is a bubble sort graph. This result enhances a result of Alahmadi et al. [Math. Meth. Appl. Sci. 39 (2016), 4856–4865]: BS is a partial cube. As a corrollary, we give the analytical expressions of the Wiener indices of bubble sort graphs.

V. S. Guba

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

2021 ◽  
Vol 28 (02) ◽  
pp. 309-318
Bo Ling ◽  
Bengong Lou ◽  
Li Ma ◽  
Xue Yu

A Cayley graph [Formula: see text] is said to be normal if [Formula: see text] is normal in [Formula: see text]. In this paper, we investigate the normality problem of the connected 11-valent symmetric Cayley graphs [Formula: see text] of finite nonabelian simple groups [Formula: see text], where the vertex stabilizer [Formula: see text] is soluble for [Formula: see text] and [Formula: see text]. We prove that either [Formula: see text] is normal or [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Further, 11-valent symmetric nonnormal Cayley graphs of [Formula: see text], [Formula: see text] and [Formula: see text] are constructed. This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind (of valency 11) was constructed by Fang, Ma and Wang in 2011.

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