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Register a new userThe Cyclic Graph of a Z-group
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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 $Z$-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $Delta(G)$ for a $Z$-group $G$.
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The cyclic graph $Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such that whose cyclic graph $Gamma(S)$ is complete, bipartite, tree, regular and a null graph, respectively. Further, we determine the clique number of $Gamma(S)$ for an arbitrary semigroup $S$. We obtain the independence number of $Gamma(S)$ if $S$ is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of $Gamma(S)$ if $S$ is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.
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 $S_n$.
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 $xH,yHin langle zHrangle$ for some $zin G$. In this article, we characterize the enhanced quotient graph of $G/H$. The graph $mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and $mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some relation between the graph $mathcal{G}_{H}(G)$ and the enhanced power graph $mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph $mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and only if $mathcal{G}(G/H)$ is complete. The graph $mathcal{G}_H(G)$ is Eulerian if and only if $|G|$ is odd if and only if $mathcal{G}(G)$ is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup $H$.
It is not known whether Thompsons group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question is whether F is graph automatic or C-graph automatic for an appropriate language class C. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
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