On the $\alpha$-spectral radius of the $k$-uniform supertrees

TL;DR

This paper investigates the maximum $oldsymbol{ extit{ extalpha}}$-spectral radius of $k$-uniform supertrees, identifying unique extremal structures based on parameters like edges, independence number, degree sequences, and matching number.

Contribution

It determines the unique supertrees with maximum $ extbf{ extalpha}$-spectral radius under various constraints, extending spectral theory to hypergraph supertrees.

Findings

Identifies supertrees with maximum $ extbf{ extalpha}$-spectral radius for given parameters.

Provides characterizations of extremal supertrees based on independence and matching numbers.

Establishes the uniqueness of these extremal supertrees.

Abstract

Let $G$ be a $k$-uniform hypergraph with vertex set $V(G)$ and edge set $E(G)$. A connected and acyclic hypergraph is called a supertree. For $0\leq\alpha<1$, the $\alpha$-spectral radius of $G$ is the largest $H$-eigenvalue of $\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ and $A(G)$ are the diagonal tensor of the degrees and the adjacency tensor of $G$, respectively. In this paper, we determine the unique supertrees with the maximum $\alpha$-spectral radius among all $k$-uniform supertrees with $m$ edges and independence number $\beta$ for $\lceil\frac{m(k-1)+1}{k}\rceil\leq\beta\leq m$, among all $k$-uniform supertrees with given degree sequences, and among all $k$-uniform supertrees with $m$ edges and matching number $\mu$ for $1\leq\mu\leq\lfloor\frac{m(k-1)+1}{k}\rfloor$, respectively.