Related papers: On $\BCI$-groups and $\CI$-groups
This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that…
A well-known conjecture of Babai states that if $G$ is any finite simple group and $X$ is a generating set for $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded by $\log|G|^c$ for some universal constant $c$. In this paper,…
The power graph of a group $G$, denoted as $P(G)$, constitutes a simple undirected graph characterized by its vertex set $G$. Specifically, vertices $a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup generated by $b$…
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of…
We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…
Let $G$ be an abelian group. The main theorem of this paper asserts that there exists a Cayley graph on $G$ with chromatic number $3$ if and only if $G$ is not of exponent $1$, $2$, or $4$. For connected Cayley graphs, we also show that…
Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp_S(G^\sigma)$.…
We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly…
For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…
A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a…
The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that…
A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition…
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical…
We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$.…
A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral,…
Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G),…
An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph $G$ is said to be $(2,3)$-orientable if there exists a friendly…
In this note we prove that all connected Cayley graphs of every finite group $Q \times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order.
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…