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For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with the spectral radius up to a scalar. For a general nonnegative tensor, we characterize the nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs., Comment: arXiv admin note: text overlap with arXiv:1707.07414

In this paper we investigate the traces of the adjacency tensor of hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and provide some perturbation results for the traces of a hypergraph with cut vertices. As applications we determine the unique hypertree with maximum Estrada index among all hypertrees with fixed number of edges and perfect matchings, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with fixed number of edges and girth $3$.

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 devote to 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 $$\Delta^{1\over k} \leq \lambda(\h)< \frac{k}{k-1}((k-1)(\Delta-1))^{1\over k}.$$ 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 Godsil's identity ({\em Matchings and walks in graphs. J. Graph Theory} 5 (1981) 285--297) on the matching polynomial of graphs.

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue.

The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.

Finding a suitable measurement matrix is an important topic in compressed sensing. Though the known random matrix, whose entries are drawn independently from a certain probability distribution, can be used as a measurement matrix and recover signal well, in most cases, we hope the measurement matrix imposed with some special structure. In this paper, based on random graph models, we show that the mixed symmetric random matrices, whose diagonal entries obey a distribution and non-diagonal entries obey another distribution, can be used to recover signal successfully with high probability., Comment: 10 pages, 3 figures. arXiv admin note: text overlap with arXiv:1212.3799

In this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of fixed order and given edge connectivity.