Perfect State Transfer on Cayley Graphs over Dihedral Groups: The Non-Normal Case

Abstract

Recently, perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group $D_n$, we show that ${\rm Cay}(D_n,S)$ cannot have PST if $n$ is odd. For even integers $n$, it is proved that if ${\rm Cay}(D_n,S)$ has PST, then $S$ is normal.

Perfect state transfer on bi-Cayley graphs over abelian groups

“…Section 4 gives necessary and sufficient conditions for an abelian bi-Cayley graph having PST (see Theorems 4.6,4.11 and 4.14). As corollaries, many known results in [11,12,35,39] can be obtained. Section 5 examines PST on Cayley graphs that are also bi-Cayley graphs.…”

“…Section 5 examines PST on Cayley graphs that are also bi-Cayley graphs. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having PST between two distinct vertices (see Example 5.1), which produces a counterexample of Theorem 11 in [12]. Concluding remarks are given in Section 6.…”

“…Using group representations, Cao et al examined perfect state transfer for Cayley graphs over abelian groups [39], dihedral groups [11,12], and semi-dihedral groups [35]. Bi-Cayley graphs are a natural generalization of Cayley graphs, and have received considerable attention during the past few years (see [19,33,40] for example).…”

State transfers in vertex complemented coronas

“…Up until now, many graphs have been proved to have or not have PST as well as PGST, including trees [5,15,18,23], Cayley graphs [3,4,[7][8][9][10]25,29,30,32,35,36], distance regular graphs [16] and some graph operations such as NEPS [11,12,25,31,33,38], coronas [1] and joins [2]. For more information, we refer the reader to [13,14,21,22,39,40].…”

The ability of UHPC prepared with unpurified sea sand and seawater to protect embedded steel against corrosion

“…China has a vast sea area with coastlines of 18,000 kilometres, and abundant sea sand reserves [1,2]. If unpurified sea sand and seawater can be used as construction sand in coastal areas, it can save freshwater resources for desalinating sea sand and reap huge economic and ecological benefits.…”