In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and determines Cayley graphs over generalized dihedral groups and generalized dicyclic groups having PST.
We show that for certain integers $n$, the problem of whether or not a Cayley digraph $Γ$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.
Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
College of Liberal Arts and Sciences, Kitasato University（北里大学文理学院）；Department of Mathematical Sciences, Faculty of Science, Shinshu University（信州大学理学部数学科学系）
A perfect code $C$ in a graph $Γ$ is an independent set of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $Γ$ is a set of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of…
Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His pr…
We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, …
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
Let $G$ be a finite abelian group written additively with identity $0$, and $Ω$ be an inverse closed generating subset of $G$ such that $0\notin Ω$. We say that $ Ω$ has the property \lq\lq{}$us$\rq\rq{} (unique summation), whenever for every $0 \neq g\in G$ if there are $s_1,s_2,s_3, s_4 \in Ω$ such that $s_1+s_2=g=s_3+s_4 $, then we have $\{s_1,s_2 \} = \{s_3,s_4 \}$. We say that a Cayley graph $Γ=Cay(G;Ω)$ is a $us$-$Cayley\ graph$, whenever $G$ is an abelian group and the generating subset $Ω$ has the property \lq\lq{}$us$\rq\rq{}. In this paper, we show that if $Γ=Cay(G;Ω)$ is a $us$-$Cay…
We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature $κ$, abelian groups are identically flat, and in the other direction we show that $κ\equiv 0$ implies the group is virtually abelian. Beyond that, $κ$ c…
