Abstract:
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.

Abstract:
Following a problem posed by Lov\'asz 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 vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0 modulo 4.

Abstract:
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.

Abstract:
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 $k\ge 4$.

Abstract:
A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A $0$-type graph can be represented as the graph $\mathrm{BCay}(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{\{h_0,g_1\} : h,g \in H, g h^{-1} \in S\}$. A bi-Cayley graph $\mathrm{BCay}(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\mathrm{BCay}(H,T),$ $\mathrm{BCay}(H,S) \cong \mathrm{BCay}(H,T)$ implies that $T = h S^\alpha$ for some $h \in H$ and $\alpha \in \mathrm{Aut}(H)$. It is also shown that every cubic connected arc-transitive $0$-type bi-Cayley graph over an abelian group is a BCI-graph.

Abstract:
A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented.

Abstract:
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.

Abstract:
Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency . 1. Introduction We assume that all graphs in this paper are finite, simple, and undirected. Let be a graph. Denote the vertex set, arc set, and full automorphism group of by , , and , respectively. A graph is called -vertex-transitive or -arc-transitive if acts transitively on or , where . is simply called vertex-transitive, arc-transitive for the case where . In particular, is called -regular if acts regularly on its arcs and then 1-regular when . Let be a finite group with identity element . For a subset of with , the Cayley graph of (with respect to ) is defined as the graph with vertex set such that are adjacent if and only if . It is easy to see that a Cayley graph has valency , and it is connected if and only if . Li proved in [1] that there are only finite number of core-free -transitive Cayley graphs of valency for and and that, with the exceptions and , every -transitive Cayley graph is a normal cover of a core-free one. It was proved in [2] that there are core-free -transitive cubic Cayley graphs up to isomorphism, and there are no core-free -regular cubic Cayley graphs. A natural problem arises. Characterize -transitive Cayley graphs, in particular, which graphs are -regular? Until now, the result about -regular graphs mainly focused constructing examples. For example, Frucht gave the first example of cubic -regular graph in [3]. After then, Conder and Praeger constructed two infinite families of cubic -regular graphs in [4]. Maru？i？ [5] and Malni？ et al. [6] constructed two infinite families of tetravalent -regular graphs. Classifying such graphs has aroused great interest. Motivated by above results and problem, we consider -regular Cayley graphs of valency 5 in this paper. A graph can be viewed as a Cayley graph of a group if and only if contains a subgroup that is isomorphic to and acts regularly on the vertex set. For convenience, we denote this regular subgroup still by . If contains a normal subgroup that is regular and isomorphic to , then is called an X-normal Cayley graph of ; if is not normal in but has a subgroup which is normal in and semiregular on with exactly two orbits, then is called an X-bi-normal Cayley graph; furthermore if , is called normal or bi-normal. Some

Abstract:
We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions.

Abstract:
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find enumeration formulae of the equivalence classes and weak equivalence classes of Cayley graphs. As an application, we find the number of weak equivalence classes of circulant graphs.