The enhanced power graph of a group G is a graph with vertex set G, where two distinct vertices x and y are adjacent if and only if there exists an element w in G such that both x and y are powers of w. In this paper, we determine the vertex connectivity of the enhanced power graph of any finite nilpotent group.
The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$.
We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively. We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$.
We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.
For a group 𝐺, the enhanced power graph of 𝐺 is a graph with vertex set 𝐺 in which two distinct vertices
x
,
y
x,y
are adjacent if and only if there exists an element 𝑤 in 𝐺 such that both 𝑥 and 𝑦 are powers of 𝑤.
The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set
G
∖
S
G\setminus S
, where 𝑆 is the set of dominating vertices of the enhanced power graph.
In this paper, we at first classify all nilpotent groups 𝐺 such that the proper enhanced power graphs are connected and calculate their diameter.
We also explicitly calculate the domination number of the proper enhanced power graphs of finite nilpotent groups.
Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.
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