Related papers: Groups all of whose undirected Cayley graphs are i…

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct…

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such…

We introduce a family of graphs that generalises the class of Cayley graphs. For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R) is the directed graph with vertex set G and an arc from x to y if and only if…

Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation),…

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.

A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley…

Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we…

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its…

For any finite abelian group $G$ and any subset $S\seq G$, we determine the connectivity of the addition Cayley graph induced by $S$ on $G$. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a…

The non-centralizer graph of a finite group $G$ is the simple graph $\Upsilon_G$ whose vertices are the elements of $G$ with two vertices $x$ and $y$ are adjacent if their centralizers are distinct. The induced subgroup of $\Upsilon_G$…

Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $\Gamma(R, G, S)$, corresponding to a multiplicative subgroup…

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…

Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…

A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient…

If $G$ is a finite group, then the spectrum $\omega(G)$ is the set of all element orders of $G$. The prime spectrum $\pi(G)$ is the set of all primes belonging to $\omega(G)$. A simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and in…

In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a…

A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$…