Related papers: Toroidal involutory Cayley graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this…

A rectangular dual of a plane graph $G$ is a contact representation of $G$ by interior-disjoint rectangles such that (i) no four rectangles share a point, and (ii) the union of all rectangles is a rectangle. In this paper, we study…

Let $R$ be a ring with unity. The \emph{idempotent graph} $G_{\text{Id}}(R)$ of a ring $R$ is an undirected simple graph whose vertices are the set of all the elements of ring $R$ and two vertices $x$ and $y$ are adjacent if and only if…

Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring $k[G]$ via those of toric rings associated to…

From the point of view of discrete geometry, the class of locally finite transitive graphs is a wide and important one. The subclass of Cayley graphs is of particular interest, as testifies the development of geometric group theory. Recall…

For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of…

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in…

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…

In this paper, we show that if $G$ is $3$-connected, then the Cartesian product of graphs $G \square H$ embeds on the torus if and only if $G$ is outer-cylindrical and $H$ is a path on two vertices, $P_2$. As a by-product of our work, we…

Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a…

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian…

Let R be a commutative ring with identity and I be an ideal of R. The cozero-divisor graph with respect to I, denoted by $\Gamma''_I(R)$, is the graph of R with vertices {x \in R -I :xR +I \not=R} and two distinct vertices $x$ and $y$ are…

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures…

Given a graph $G$ with vertex set $\{1,\ldots,n\}$, we can project the graphical arrangement of $G$ to an $(n-1)$-dimensional torus to obtain a toric hyperplane arrangement. Adams, Defant, and Striker constructed a toric combinatorial…

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…

In this note we study embeddings of Cayley graphs of right groups on surfaces. We characterize those right groups which have a toroidal but no planar Cayley graph, such that the generating system of the right group has a minimal generating…

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set ${{a,b}:a,b\in R, a-b\in R^\times}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is…

The intersection graph of ideals associated with a commutative unitary ring $R$ is the graph $G(R)$ whose vertices all non-trivial ideals of $R$ and there exists an edge between distinct vertices if and only if the intersection of them is…