CONNECTED COMPONENTS IN THE INVARIABLY GENERATING GRAPH OF A FINITE GROUP

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

The Non-Commuting, Non-Generating Graph of a Nilpotent Group

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

The invariably generating graph of the alternating and symmetric groups

Given a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.

The virtually generating graph of a profinite group

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

GenGraph: a python module for the simple generation and manipulation of genome graphs

Background As sequencing technology improves, the concept of a single reference genome is becoming increasingly restricting. In the case of Mycobacterium tuberculosis, one must often choose between using a genome that is closely related to the isolate, or one that is annotated in detail. One promising solution to this problem is through the graph based representation of collections of genomes as a single genome graph. Though there are currently a handful of tools that can create genome graphs and have demonstrated the advantages of this new paradigm, there still exists a need for flexible tools that can be used by researchers to overcome challenges in genomics studies. Results We present GenGraph, a Python toolkit and accompanying modules that use existing multiple sequence alignment tools to create genome graphs. Python is one of the most popular coding languages for the biological sciences, and by providing these tools, GenGraph makes it easier to experiment and develop new tools that utilise genome graphs. The conceptual model used is highly intuitive, and as much as possible the graph structure represents the biological relationship between the genomes. This design means that users will quickly be able to start creating genome graphs and using them in their own projects. We outline the methods used in the generation of the graphs, and give some examples of how the created graphs may be used. GenGraph utilises existing file formats and methods in the generation of these graphs, allowing graphs to be visualised and imported with widely used applications, including Cytoscape, R, and Java Script. Conclusions GenGraph provides a set of tools for generating graph based representations of sets of sequences with a simple conceptual model, written in the widely used coding language Python, and publicly available on Github.

THE GENERATING GRAPH OF INFINITE ABELIAN GROUPS

For a group $G$, let $\unicode[STIX]{x1D6E4}(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Let $\unicode[STIX]{x1D6E4}^{\ast }(G)$ be the subgraph of $\unicode[STIX]{x1D6E4}(G)$ that is induced by all the vertices of $\unicode[STIX]{x1D6E4}(G)$ that are not isolated. We prove that if $G$ is a 2-generated noncyclic abelian group, then $\unicode[STIX]{x1D6E4}^{\ast }(G)$ is connected. Moreover, $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$ if the torsion subgroup of $G$ is nontrivial and $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $\unicode[STIX]{x1D6E4}^{\ast }(F)$ is connected and we deduce from $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$.

GenGraph: a python module for the simple generation and manipulation of genome graphs

BackgroundAs sequencing technology improves, the concept of a single reference genome is becoming increasingly restricting. In the case of Mycobacterium tuberculosis, one must often choose between using a genome that is closely related to the isolate, or one that is annotated in detail. One promising solution to this problem is through the graph based representation of collections of genomes as a single genome graph. Though there are currently a handful of tools that can create genome graphs and have demonstrated the advantages of this new paradigm, there still exists a need for flexible tools that can be used by researchers to overcome challenges in genomics studies.ResultsWe present the GenGraph toolkit, a tool that uses existing multiple sequence alignment tools to create genome graphs. It is written in Python, one of the most popular coding languages for the biological sciences, and creates the genome graphs as Python NetworkX graph objects. The conceptual model is highly intuitive, and as much as possible represents the biological relationship between the genomes. This design means that users will quickly be able to start creating genome graphs and using them in their own projects.We outline the methods used in the generation of the graphs, and give some examples of how the created graphs may be used. GenGraph utilises existing file formats and methods in the generation of these graphs, allowing graphs to be visualised and imported with widely used applications, including Cytoscape, R, and Java Script.ConclusionGenGraph provides a set of tools for generating graph based representations of sets of sequences with a simple conceptual model in a widely used coding language. It is publicly available on Github (https://github.com/jambler24/GenGraph).