arXiv subjects
Publications and source records attributed to Mohammad A. Iranmanesh.
A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order $6p$ are classified for each prime $p$.
A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ which leaves invariant each of the $G$-orbits for the induced action on $Ω\times Ω$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the Monster $\mathbb{M}$. These are the first known examples of insoluble totally $2$-closed groups. As a by-product of our methods, we develop several tools for studying $2$-closures of transitive permutation groups -- a vital tool in the study of representations of finite groups as automorphism groups of digraphs. We also prove a dual to a 1939 theorem of Frucht from Algebraic Graph Theory.
Let $G$ be a finite non-abelian group and $m=|G|/|Z(G)|$. In this paper we investigate $m$-centralizer group $G$ with cyclic center and we will prove that if $G$ is a finite non-abelian $m$-centralizer $CA$-group, then there exists an integer $r>1$ such that $m=2^r.$ It is also prove that if $G$ is an $m$-centralizer non-abelian finite group which is not a $CA$-group and its derived subgroup $G'$ is of order 2, then there exists an integer $s>1$ such that $m=2^{2s}.$
The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.
Let $Σ=(Γ, σ)$ is a signed graph(or sigraph in short), where $Γ$ is a underlying graph of $Σ$ and $σ:E\longrightarrow \{+, -\}$ is a function. Consider $Γ=Cay(\mathbb{Z}_{p_{1}}\times \mathbb{Z}_{p_{1}^{α_{1}}p_{2}^{α_{2}} \ldots p_{k}^{α_{k}}}, Φ)$, where all $p_{1}, p_{2}, \ldots, p_{k}$ are distinct prime factors and $Φ=φ_{p_{1}}\timesφ_{p_{1}^{α_{1}}p_{2}^{α_{2}} \ldots p_{k}^{α_{k}}}$. For any positive integer $n$, $φ_{n}=\{\ell| 1\leq \ell<n, \gcd(\ell, n)=1\}$. Motivated by \cite{s14}, we will investigate balancing in $Σ$ and $L(Σ)$, clusterability and sign-compatibility of $Σ$.
Using the domination parameters of Cayley graphs constructed out of $\mathbb{Z}_{p}\times \mathbb{Z}_{m}$, where $m\in\{p^α, p^αq^β, p^αq^βr^γ\},$ in this paper we are discussing about the total and connected domination number and diameter of these Cayley graphs.
Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. A bi-Cayley graph $\BCay(G,S)$ is called a $\BCI$-graph if for any bi-Cayley graph $\BCay(G,T)$, whenever $\BCay(G,S)\cong\BCay(G,T)$ we have $T=gS^σ$ for some $g\in G$ and $σ\in\Aut(G).$ A group $G$ is called a $\BCI$-group if every bi-Cayley graph of $G$ is a $\BCI$-graph. In this paper, we showed that every $\BCI$-group is a $\CI$-group, which gives a positive answer to a conjecture proposed by Arezoomand and Taeri in \cite{arezoomand1}. Also we proved that there is no any non-Abelian $4$-$\BCI$-simple group. In addition all $\BCI$-groups of order $2p$, $p$ a prime, are characterized.
In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$.
The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.
Let $p$ and $q$ be distinct odd primes. Let $Γ=(V(Γ), E(Γ))$ be a non-Cayley vertex-transitive graph of order $pq.$ Let $G\leq \Aut(Γ)$ acts primitively on the vertex set $V(Γ)$. In this paper, we show that $G$ is uniprimitive which is primitive but not 2-transitive and we obtain some information about $p, q$ and the minimality of the Socle $T=\soc(G).$
Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$.
The divisibility graph $D(G)$ for a finite group $G$ is a graph with vertex set $cs~(G)\setminus\{1\}$ where $cs~(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. In this paper we will find $D(G)$ where $G$ is a simple Zassenhaus group.
Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph $\Ga(X)$ and the prime vertex graph $Δ(X)$, for a set $X$ of positive integers, we define the bipartite divisor graph $B(X)$, and show that many of these connections flow naturally from properties of $B(X)$. In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as $B(X)$ for some $X$. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of $B(X)$, for one $Γ(X)$ or $Δ(X)$ to contain a complete subgraph of size 3 or 4.