The non-$\mathfrak{F}$ graph of a finite group

Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.
Given a class of groups $\mathfrak{F}$ and a group $G$, we consider the graph whose vertices are the elements of $G$, and there is an edge between two vertices $g,h \in G$, if $\langle g,h\rangle \not \in \mathfrak F$. The subgraph induced by the non-isolated vertices is the non-$\mathfrak F$ graph of $G$. This object is a generalization of some known graphs  (e.g. the non-commuting graph defined by Paul Erdös) previously studied with ad-hoc techniques which we try to put in a general framework. We investigate mainly the set of isolated vertices, which sometimes is a subgroup with an algebraic meaning and some connectivity properties. We apply these results to various notable classes.



Date and Venue

Start Date
Online seminar
End Date


Daniele Nemmi

Speaker's Institution

Università di Padova



Algebra, Combinatorics and Number Theory