The ordering of hypertrees and unicyclic hypergraphs by the traces of ${\mathcal{A}}_{\alpha }$-tensor

Abstract

For a real number $\alpha \in [0,1]$ and a $k$-uniform hypergraph $\mathcal{H}$, ${\mathcal{A}}_{\alpha }(\mathcal{H})=\alpha \mathcal{D}(\mathcal{H})+(1-\alpha )\mathcal{A}(\mathcal{H})$ is called the ${\mathcal{A}}_{\alpha }$-tensor of $\mathcal{H}$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree tensor and adjacency tensor of $\mathcal{H}$, respectively. The sum of the $d$-th powers of all eigenvalues of ${\mathcal{A}}_{\alpha }(\mathcal{H})$ is called the $d$-th order ${\mathcal{A}}_{\alpha }$-spectral moment of $\mathcal{H}$, which is equal to the $d$-th order trace of ${\mathcal{A}}_{\alpha }(\mathcal{H})$. In this paper, some hypergraphs are ordered lexicographically by their ${\mathcal{A}}_{\alpha }$-spectral moments in non-decreasing order. The first, the second, the last and the second last hypergraphs among all $k$-uniform linear unicyclic hypergraphs and hypertrees are characterized, respectively. We give the first and the last hypergraphs among all $k$-uniform linear unicyclic hypergraphs with given grith, and characterize the last hypertree among all $k$-uniform hypertrees with given diameter. Furthermore, we determine some extreme values of the ${\mathcal{A}}_{\alpha }$-spectral moments for hypertrees and linear unicyclic hypergraphs, respectively.