We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of...We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges （2d ≥ k ≥ 3）, we show that its largest （signless） Laplacian Z-eigenvalue is d.展开更多
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
With the rapid development of the Internet, recent years have seen the explosive growth of social media. This brings great challenges in performing efficient and accurate image retrieval on a large scale. Recent work ...With the rapid development of the Internet, recent years have seen the explosive growth of social media. This brings great challenges in performing efficient and accurate image retrieval on a large scale. Recent work shows that using hashing methods to embed high-dimensional image features and tag information into Hamming space provides a powerful way to index large collections of social images. By learning hash codes through a spectral graph partitioning algorithm, spectral hashing(SH) has shown promising performance among various hashing approaches. However, it is incomplete to model the relations among images only by pairwise simple graphs which ignore the relationship in a higher order. In this paper, we utilize a probabilistic hypergraph model to learn hash codes for social image retrieval. A probabilistic hypergraph model offers a higher order repre-sentation among social images by connecting more than two images in one hyperedge. Unlike a normal hypergraph model, a probabilistic hypergraph model considers not only the grouping information, but also the similarities between vertices in hy-peredges. Experiments on Flickr image datasets verify the performance of our proposed approach.展开更多
We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3,...We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3, we show that the largest H-eigenvalue of its adjacency tensor is （（1 ＋ √-5）/2）2/k when = 3 and ）λ（A） = 31/k when g = 4, respectively. For the case of l ≥ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval （2, 3） when l≥ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.展开更多
基金Acknowledgements Foundation of China This work was supported by the National Natural Science （Grant Nos. 11371109, 11426075）, the Natural Science Foundation of tteilongjiang Province （No. QC2014C001）, ：and the Fundamental Research Funds for the Central Universities
文摘We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges （2d ≥ k ≥ 3）, we show that its largest （signless） Laplacian Z-eigenvalue is d.
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Project supported by the National Basic Research Program(973)of China(No.2012CB316400)
文摘With the rapid development of the Internet, recent years have seen the explosive growth of social media. This brings great challenges in performing efficient and accurate image retrieval on a large scale. Recent work shows that using hashing methods to embed high-dimensional image features and tag information into Hamming space provides a powerful way to index large collections of social images. By learning hash codes through a spectral graph partitioning algorithm, spectral hashing(SH) has shown promising performance among various hashing approaches. However, it is incomplete to model the relations among images only by pairwise simple graphs which ignore the relationship in a higher order. In this paper, we utilize a probabilistic hypergraph model to learn hash codes for social image retrieval. A probabilistic hypergraph model offers a higher order repre-sentation among social images by connecting more than two images in one hyperedge. Unlike a normal hypergraph model, a probabilistic hypergraph model considers not only the grouping information, but also the similarities between vertices in hy-peredges. Experiments on Flickr image datasets verify the performance of our proposed approach.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271221） and the Specialized Research Fund for State Key Laboratories.
文摘We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3, we show that the largest H-eigenvalue of its adjacency tensor is （（1 ＋ √-5）/2）2/k when = 3 and ）λ（A） = 31/k when g = 4, respectively. For the case of l ≥ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval （2, 3） when l≥ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.
