Agelos Georgakopoulos
We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.
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We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.
A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.
For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified $\mathcal{G}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class $\mathcal{G}_3$ is characterized. As an application, the classification of $\mathcal{G}_k$ is obtained again, where $k\ge 4$.
A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.
Following a problem posed by Lovász in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=\la a,b| a^2=1, b^s=1, (ab)^3=1, etc. \ra$ generated by an involution $a$ and an element $b$ of order $s\geq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two ve…
In this paper, rough approximations of Cayley graphs are studied and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition called pseudo-Cayley graphs containing Cayley graphs is proposed. Rough approximation is expanded to pseudo-Cayley graphs. Also, rough vertex pseudo-Cayley graphs and rough pseudo-Cayley graphs are introduced. Some theorems are provided, form which some properties such as connectivity and optimal connectivity are derived. This approach opens a new research field in sciences such as data networks.
The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed subset of $G$. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.
Prime Square Order Cayley Graph of Cyclic Groups of Particular Valency
研究素数平方乘积的循环群生成的Cayley图结构，分析连通性、欧拉性、哈密顿性及关键图参数。
On Some Bi-Cayley Graphs over Cyclic Groups of Order $p^2 q^2$ and Related Extensions
本文研究了阶为$p^2 q^2$的循环群上的双Cayley图的结构和性质，证明其连通性、双正则性及环长为三，并扩展了相关结果到更一般的有限群情况。
We prove that if the Cayley graph of a finitely generated group enjoys the property L_delta then the group is almost convex and has a sub-cubic isoperimetric function.
Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups
我们研究随机右角Coxeter群的几何，解决方形渗透连通性猜想，并证明在广泛参数范围内（含p=1/2）其具有唯一的立方粗中值结构，即立方粗刚性典型成立。
Connective constants of Grigorchuk graphs
本文研究Grigorchuk群的Cayley图的连接常数，证明其下界大于黄金分割数φ，扩展了关于无限顶点传递立方图连接常数的猜想，并分析了轨道Schreier图的标签序列出现比例。
Integral Cayley graphs of symmetric groups on transpositions
本研究针对对称群对换子集的整Cayley图问题，证明树图对应整Cayley图当且仅当为星图，提出广义完全多部图对应整图的结论及反向猜想，并在圈、三次图等图类中验证猜想有效性。
Generating the symmetric group by three prefix reversals
本文研究了通过三个前缀翻转生成对称群的问题，确定了部分生成集并探讨了立方煎饼图的直径和圈长。
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