On the $\hat{I}\pm$-spectral radius of the $k$-uniform 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 $0\le \hat{I}\pm <1$, the $\hat{I}\pm$-spectral radius of $G$ is the largest $H$-eigenvalue of $\hat{I}\pm D(G)+(1-\hat{I}\pm )A(G)$, 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 $\hat{I}\pm$-spectral radius among all $k$-uniform supertrees with $m$ edges and independence number ${\hat{I}}^2$ for $\lceil \frac{m(k-1)+1}{k}\rceil \le {\hat{I}}^2\le m$, among all $k$-uniform supertrees with given degree sequences, and among all $k$-uniform supertrees with $m$ edges and matching number $\hat{I}¼$ for $1\le \hat{I}¼\le \lfloor \frac{m(k-1)+1}{k}\rfloor$, respectively.