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It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and matching polynomials to determine the algebraic multiplicity of the spectral radius of a uniform hypertree.

We show how all eigenvalues of a power hypergraph $G^{(k)}$ can be generated from the eigenvalues of signed subgraphs of the underlying graph $G$. This fixes an incorrect statement in the case of power hypergraphs from [Linear Algebra and its Applications, 590:243-257, 2020].

Uniform hypergraphs have a natural one-to-one correspondence to tensors. In this paper, we investigate the Estrada index and subgraph centrality of an $m$-uniform hypergraph $\mathcal{H}$ via the adjacency tensor. We establish some bounds for the Estrada index and give expressions of the subgraph centrality in terms of graph parameters of the multi-digraphs associated with $\mathcal{H}$. When $\mathcal{H}$ is $2$-uniform, the above Estrada index and subgraph centrality are the Estrada index and subgraph centrality of a graph.

For a simple, undirected and connected graph $G$, $D_��(G) = ��Tr(G) + (1-��) D(G)$ is called the $��$-distance matrix of $G$, where $��\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $��$-distance energy of $G$ was defined based on the spectra of $D_��(G)$. In this paper, we define the $��$-distance Estrada index of $G$ in terms of the eigenvalues of $D_��(G)$. And we give some bounds on the spectral radius of $D_��(G)$, $��$-distance energy and $��$-distance Estrada index of $G$.

In this paper, we give some bounds for principal eigenvector and spectral radius of connected uniform hypergraphs in terms of vertex degrees, the diameter, and the number of vertices and edges.

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a}, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory, Ser. B 97 (2007) 483-486] to hypergraphs. Applying these bounds, we also obtain a lower bound for $\Delta+\mu$.

Suppose F is a field of characteristic not 2 or 3. A characterization is given for all additive maps, on the algebra of all n × n matrices over F. which preserve Moore -Penrose(M-P) Inverses of matrices.

In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.