arXiv subjects

Emanuele Pacifici

Publications and source records attributed to Emanuele Pacifici.

20 records · Page 2Linked to original sources

arXiv2016

Groups whose character degree graph has diameter three

Let $G$ be a finite group, and let $Δ(G)$ denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of $G$. It is well known that, whenever $Δ(G)$ is connected, the diameter of $Δ(G)$ is at most $3$. In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.

math.GR↗

arXiv2013

Conjugacy classes of finite groups and graph regularity

Given a finite group $G$, denote by $Γ(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $Γ(G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if $Γ(G)$ is a $k$-regular graph with $k\geq 1$, then $Γ(G)$ is a complete graph with $k+1$ vertices.

math.GR↗