The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smal...The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.展开更多
Let G be a simple connected graph of order n ≥ 6. The third edge-connectivity of G is defined as the minimum cardinality over all the sets of edges, if any, whose deletion disconnects G and every component of the res...Let G be a simple connected graph of order n ≥ 6. The third edge-connectivity of G is defined as the minimum cardinality over all the sets of edges, if any, whose deletion disconnects G and every component of the resulting graph has at least 3 vertices. In this paper, we first characterize those graphs whose third-edge connectivity is well defined,then establish the tight upper bound for the third edge-connectivity.展开更多
An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the c...An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ<SUB> m </SUB>= min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ<SUB> m </SUB>≤ ξ<SUB> m </SUB>. The upper bound of λ<SUB> m </SUB>is sharp.展开更多
It is shown that the lower bound on the maximum genus of a 3-edge connected loopless graph is at least one-third of its cycle rank. Moreover, this lower bound is tight. There are infinitely such graphs attaining the b...It is shown that the lower bound on the maximum genus of a 3-edge connected loopless graph is at least one-third of its cycle rank. Moreover, this lower bound is tight. There are infinitely such graphs attaining the bound.展开更多
Spectral computed tomography(CT) based on photon counting detectors(PCDs) is a well-researched topic in the field of X-ray imaging. When PCD is applied in a spectral CT system, the PCD energy thresholds must be carefu...Spectral computed tomography(CT) based on photon counting detectors(PCDs) is a well-researched topic in the field of X-ray imaging. When PCD is applied in a spectral CT system, the PCD energy thresholds must be carefully selected, especially for K-edge imaging, which is an important spectral CT application. This paper presents a threshold selection method that yields better-quality images in K-edge imaging. The main idea is to optimize the energy thresholds ray-by-ray according to the targeted component coefficients, followed by obtaining an overall optimal energy threshold by frequency voting. A low-dose pre-scan is used in practical implementations to estimate the line integrals of the component coefficients for the basis functions. The variance of the decomposed component coefficients is then minimized using the Cramer–Rao lower bound method with respect to the energy thresholds. The optimal energy thresholds are then used to take a full scan and gain better image reconstruction with less noise than would be given by a full scan using the non-optimal energy thresholds. Simulations and practical experiments on imaging iodine and gadolinium solutions, which are commonly used as contrast agents in medical applications, were used to validate the method. The noise was significantly reduced with the same dose relative to the non-optimal energy thresholds in both simulations and in practical experiments.展开更多
A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge ...A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph C is denoted by Xs＇8（G）. In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with △（G） ≥ 4, respectively.展开更多
文摘The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.
基金This work was supported by Zhejiang Provincial Natural Science Foundation of China(Grant No.102055)the Natural Science Foundation of Zhejiang Normal UniversityThe second author was supported by the National Natural Science Foundation of China(Grant No.19971056).
文摘Let G be a simple connected graph of order n ≥ 6. The third edge-connectivity of G is defined as the minimum cardinality over all the sets of edges, if any, whose deletion disconnects G and every component of the resulting graph has at least 3 vertices. In this paper, we first characterize those graphs whose third-edge connectivity is well defined,then establish the tight upper bound for the third edge-connectivity.
基金National Natural Science Foundation of China (Grant No.10271105) and Doctoral Fund of Zhangzhou Normal College.
文摘An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ<SUB> m </SUB>= min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ<SUB> m </SUB>≤ ξ<SUB> m </SUB>. The upper bound of λ<SUB> m </SUB>is sharp.
文摘It is shown that the lower bound on the maximum genus of a 3-edge connected loopless graph is at least one-third of its cycle rank. Moreover, this lower bound is tight. There are infinitely such graphs attaining the bound.
基金supported by Grants from National key research and development program(No.2016YFF0101304)the National Natural Science Foundation of China(Nos.61771279,11435007)
文摘Spectral computed tomography(CT) based on photon counting detectors(PCDs) is a well-researched topic in the field of X-ray imaging. When PCD is applied in a spectral CT system, the PCD energy thresholds must be carefully selected, especially for K-edge imaging, which is an important spectral CT application. This paper presents a threshold selection method that yields better-quality images in K-edge imaging. The main idea is to optimize the energy thresholds ray-by-ray according to the targeted component coefficients, followed by obtaining an overall optimal energy threshold by frequency voting. A low-dose pre-scan is used in practical implementations to estimate the line integrals of the component coefficients for the basis functions. The variance of the decomposed component coefficients is then minimized using the Cramer–Rao lower bound method with respect to the energy thresholds. The optimal energy thresholds are then used to take a full scan and gain better image reconstruction with less noise than would be given by a full scan using the non-optimal energy thresholds. Simulations and practical experiments on imaging iodine and gadolinium solutions, which are commonly used as contrast agents in medical applications, were used to validate the method. The noise was significantly reduced with the same dose relative to the non-optimal energy thresholds in both simulations and in practical experiments.
基金Supported by the National Natural Science Foundation of China(10971198)the Zhejiang Natural Science Foundation of China(Z6110786)
文摘A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph C is denoted by Xs＇8（G）. In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with △（G） ≥ 4, respectively.
