Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEstrada index of hypergraphs via eigenvalues of tensors
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A uniform hypergraph $mathcal{H}$ is corresponding to an adjacency tensor $mathcal{A}_mathcal{H}$. We define an Estrada index of $mathcal{H}$ by using all the eigenvalues $lambda_1,dots,lambda_k$ of $mathcal{A}_mathcal{H}$ as $sum_{i=1}^k e^{lambda_i}$. The bounds for the Estrada indices of uniform hypergraphs are given. And we characterize the Estrada indices of $m$-uniform hypergraphs whose spectra of the adjacency tensors are $m$-symmetric. Specially, we characterize the Estrada indices of uniform hyperstars.
rate research
Read More
Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.
For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor ${cal A}=(a_{i_1cdots i_r}^{j_1cdots j_s}) in ({mathbb R}^n)^rtimes ({mathbb R}^m)^s$ is called partially symmetric if it is invariant under any permutation on the lower $r$ indexes and any permutation on the upper $s$ indexes. Such partially symmetric rectangular tensor arises naturally in studying directed hypergraphs. Ling and Qi [Front. Math. China, 2013] first studied the $(p,q)$-spectral radius (or singular values) and proved a Perron-Fronbenius theorem for such tensors when both $p,q geq r+s$. We improved their results by extending to all $(p,q)$ satisfying $frac{r}{p} +frac{s}{q}leq 1$. We also proved the Perron-Fronbenius theorem for general nonnegative $(r,s)$-order $(n,m)$-dimensional rectangular tensors when $frac{r}{p}+frac{s}{q}>1$. We essentially showed that this is best possible without additional conditions on $cal A$. Finally, we applied these results to study the $(p,q)$-spectral radius of $(r,s)$-uniform directed hypergraphs.
For a nonnegative weakly irreducible tensor $mathcal{A}$, we give some characterizations of the spectral radius of $mathcal{A}$, by using the digraph of tensors. As applications, some bounds on the spectral radius of the adjacency tensor and the signless Laplacian tensor of the $k$-uniform hypergraphs are shown.
Considering $ntimes ntimes n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($Omega_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $Omega_{n}$ with some positive diagonals, and the polytope ($Delta_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $Omega_{n}$.
By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
