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Let $\h$ be a connected $k$-graph with maximum degree ${\Delta}\geq 2$ and let $\mu(\h, x)$ be the matching polynomial of $\h$. In this paper, we focus on studying the distribution of zeros of the matching polynomials of $k$-graphs. We prove that the zeros (with multiplicities) of $\mu(\h, x)$ are invariant under a rotation of an angle $2\pi/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. Let $\lambda(\h)$ denote the maximum modulus of all zeros of $\mu(\h, x)$. We show that $\lambda(\h)$ is a simple root of $\mu(\h, x)$ and $$\Delta^{1\over k} \leq \lambda(\h)< \frac{k}{k-1}\big((k-1)(\Delta-1)\big)^{1\over k}.$$ To achieve these, we introduce the path tree $\T(\h,u)$ of $\h$ with respect to a vertex $u$ of $\h$, which is a $k$-tree, and prove that $$\frac{\mu(\h-u,x)}{\mu(\h, x)} = \frac{\mu(\T(\h,u)-u,x) }{\mu(\T(\h,u),x)},$$ which generalizes the celebrated Godsil's identity on the matching polynomial of graphs.

Let $T$ be a $k$-tree equipped with a weighting function $\w: V(T)\cup E(T)\rightarrow \C$, where $k \geq 3$. The weighted matching polynomial of the weighted $k$-tree $(T,\w)$ is defined to be $$ \mu(T,\w,x)= \sum_{M \in \mathcal{M}(T)}(-1)^{|M|}\prod_{e \in E(M)}\mathbf{w}(e)^k \prod_{v \in V(T)\backslash V(M)}(x-\w(v)), $$ where $\mathcal{M}(T)$ denotes the set of matchings (including empty set) of $T$. In this paper, we investigate the eigenvalues of the adjacency tensor $\A(T,\w)$ of the weighted $k$-tree $(T,\w)$. The main result provides that $\w(v)$ is an eigenvalue of $\A(T,\w)$ for every $v\in V(T)$, and if $\lambda\neq \w(v)$ for every $v\in V(T)$, then $\lambda$ is an eigenvalue of $\A(T,\w)$ if and only if there exists a subtree $T'$ of $T$ such that $\lambda$ is a root of $\mu(T',\w,x)$. Moreover, the spectral radius of $\A(T,\w)$ is equal to the largest root of $\mu(T,\w,x)$ when $\w$ is real and nonnegative. The result extends a work by Clark and Cooper ({\em On the adjacency spectra of hypertrees, Electron. J. Combin., 25 (2)(2018) $\#$P2.48}) to weighted $k$-trees. As applications, two analogues of the above work for the Laplacian and the signless Laplacian tensors of $k$-trees are obtained.

Let $H$ be a connected $m$-uniform hypergraph, and let $\mathcal{A}(H)$ be the adjacency tensor of $H$ whose spectrum is simply called the spectrum of $H$. Let $s(H)$ denote the number of eigenvectors of $\mathcal{A}(H)$ associated with the spectral radius, and $c(H)$ denote the number of eigenvalues of $\mathcal{A}(H)$ with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of $H$. Let $\bar{H}$ be a $k$-fold covering of $H$ which can be obtained from some permutation assignment in the symmetric group $\mathbf{S}_k$ on $H$. In this paper, we first characterize the connectedness of $\bar{H}$ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of $H$ and that of $\bar{H}$. By applying module theory and group representation, if $\bar{H}$ is connected, we prove that $s(H) \mid s(\bar{H})$ and $c(H) \mid c(\bar{H})$. In particular, when $\bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $\bar{H}$ contains the spectrum of $H$ and the spectrum of a signed hypergraph with $H$ as underlying hypergraph; if $m$ is odd, we give an explicit formula for $s(\bar{H})$. We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples.

The Laplacian matching polynomial of a graph $G$, denoted by $\mathscr{L\hspace{-0.7mm}M}(G,x)$, is a new graph polynomial whose all roots are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let $G$ be a connected graph. We show that $0$ is a root of $\mathscr{L\hspace{-0.7mm}M}(G, x)$ if and only if $G$ is a tree. We prove that the number of distinct positive zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ is at least equal to the length of the longest path in $G$. It is also established that the zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and $\mathscr{L\hspace{-0.7mm}M}(G-e,x)$ interlace for each edge $e$ of $G$. Using the path-tree of $G$, we present a linear algebraic approach to investigate the largest zero of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ and particularly to give tight upper and lower bounds on it.

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal non-odd-transversal if it is non-odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first eigenvectors of $G$. In this paper we give some numerical and structural properties about the first eigenvectors of $G$ which contains an odd-bipartite branch, and investigate how the least eigenvalue of $G$ changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs.

Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over $\mathbb{Z}_m$ or $\mathbb{Z}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of $G$.