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The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with trivial centre and diameter 6, the previously largest known diameter for an infinite family was 5 for $S_n$.
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(
For a group $G$, we define a graph $Delta(G)$ by letting $G^{#} = G setminus { 1 }$ be the set of vertices and by drawing an edge between distinct elements $x,yin G^{#}$ if and only if the subgroup $langle x,yrangle$ is cyclic. Recall that a $Z$-grou
The commuting graph of a group G, denoted by Gamma(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Z_m be the commutative ring of equivalence
We count the finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Gamma(H)$, we consider the number of vertices of $Gamma(H)$ to be the size of
For a finite group $G$ with a normal subgroup $H$, the enhanced quotient graph of $G/H$, denoted by $mathcal{G}_{H}(G),$ is the graph with vertex set $V=(Gbackslash H)cup {e}$ and two vertices $x$ and $y$ are edge connected if $xH = yH$ or $xH,yHin l