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

TL;DR

This paper investigates the spectral properties of hypergraphs using the $ ext{A}_ ext{alpha}$-tensor, characterizing extremal hypergraphs in various classes based on spectral moments and traces.

Contribution

It introduces a spectral ordering of hypergraphs via $ ext{A}_ ext{alpha}$-spectral moments and characterizes extremal hypergraphs within specific classes.

Findings

Identified the first, last, second, and second last hypergraphs in spectral order among certain classes.

Characterized extremal hypergraphs with given girth and diameter.

Determined bounds for spectral moments in hypertrees and unicyclic hypergraphs.

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.