The extremal spectral radii of kk-uniform supertrees

In this paper, we study some extremal problems of three kinds of spectral radii of $k$-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence $Q$-spectral radius). We call a connected and acyclic $k$-uniform hypergraph a supertree. We introduce the operation of "moving edges" for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar $\mathcal{S}_{n,k}$ attains uniquely the maximum spectral radius among all $k$-uniform supertrees on $n$ vertices. We also determine the unique $k$-uniform supertree on $n$ vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path $\mathcal{P}_{n,k}$ attains uniquely the minimum spectral radius among all $k$-th power hypertrees of $n$ vertices. Some bounds on the incidence $Q$-spectral radius are given. The relation between the incidence $Q$-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed

On a relationship between the characteristic and matching polynomials of a uniform hypertree

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with characteristic polynomial $\phi_T(\lambda)$ and matching polynomial $\varphi_T(\lambda)$, then $\phi_T(\lambda)=\varphi_T(\lambda).$ More generally, suppose $\mathcal{T}$ is an $r$-tree of size $m$ with $r\geq2$. In this paper, we extend the above classical relationship to $r$-trees and establish that $$\phi_{\mathcal{T}}(\lambda)=\prod_{H \sqsubseteq \mathcal{T}}\varphi_{H}(\lambda)^{a_{H}},$$ where the product is over all connected subgraphs $H$ of $\mathcal{T}$, and the exponent $a_{H}$ of the factor $\varphi_{H}(\lambda)$ can be written as $$a_H=b^{m-e(H)-|\partial(H)|}c^{e(H)}(b-c)^{|\partial(H)|},$$ where $e(H)$ is the size of $H$, $\partial(H)$ is the boundary of $H$, and $b=(r-1)^{r-1}, c=r^{r-2}$. In particular, for $r=2$, the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark-Cooper [{\em Electron. J. Combin.}, 2018] and show that for any subgraph $H$ of an $r$-tree $\mathcal{T}$ with $r\geq3$, $\varphi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, and additionally $\phi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, if either $r\geq 4$ or $H$ is connected when $r=3$. Moreover, a counterexample is given for the case when $H$ is a disconnected subgraph of a 3-tree.Comment: 36 pages, 4 figure

Coastal Inundation from Sea Level Rise and Typhoon Maemi

Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchive

Efficient vanishing point detection method in unstructured road environments based on dark channel prior

Vanishing point detection is a key technique in the fields such as road detection, camera calibration and visual navigation. This study presents a new vanishing point detection method, which delivers efficiency by using a dark channel prior‐based segmentation method and an adaptive straight lines search mechanism in the road region. First, the dark channel prior information is used to segment the image into a series of regions. Then the straight lines are extracted from the region contours, and the straight lines in the road region are estimated by a vertical envelope and a perspective quadrilateral constraint. The vertical envelope roughly divides the whole image into sky region, vertical region and road region. The perspective quadrilateral constraint, as the authors defined herein, eliminates the vertical lines interference inside the road region to extract the approximate straight lines in the road region. Finally, the vanishing point is estimated by the meanshift clustering method, which are computed based on the proposed grouping strategies and the intersection principles. Experiments have been conducted with a large number of road images under different environmental conditions, and the results demonstrate that the authors’ proposed algorithm can estimate vanishing point accurately and efficiently in unstructured road scenes