The definitions of the third-order elastic,piezoelectric,and dielectric constants and the properties of the associated tensors are discussed.Based on the energy conservation and coordinate transformation,the relations...The definitions of the third-order elastic,piezoelectric,and dielectric constants and the properties of the associated tensors are discussed.Based on the energy conservation and coordinate transformation,the relations among the third-order constants are obtained.Furthermore,the relations among the third-order elastic,piezoelectric,and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials.It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups,while all third-order dielectric constants are 0 for 16 kinds of point groups as well as isotropic material.The reason is that some of the point groups are centrally symmetric,and the other point groups are high symmetry.These results provide the foundation to measure these constants,to choose material,and to research nonlinear problems.Moreover,these results are helpful not only for the study of nonlinear elastic and piezoelectric problems,but also for the research on flexoelectric effects and size effects.展开更多
Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of...Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of G.The least H-eigenvalue of A(G)refers to the least real eigenvalue of A(G)associated with a real eigenvector.In this paper,we obtain a perturbation result on the least H-eigenvalue of A(G)when a branch of G attached at one vertex is relocated to another vertex,and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.展开更多
In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find o...In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids.We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.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
A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hy...A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.展开更多
The k-uniform s-hypertree G = (V, E) is an s-hypergraph, where 1 ≤ s ≤ k - 1, and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some propert...The k-uniform s-hypertree G = (V, E) is an s-hypergraph, where 1 ≤ s ≤ k - 1, and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree A. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just (△S/k).展开更多
Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(...Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.展开更多
The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph.It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph,a...The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph.It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph,and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph.In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius,where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11872186 and11272126)the Fundamental Research Funds for the Central Universities(No.HUST:2016JCTD114)
文摘The definitions of the third-order elastic,piezoelectric,and dielectric constants and the properties of the associated tensors are discussed.Based on the energy conservation and coordinate transformation,the relations among the third-order constants are obtained.Furthermore,the relations among the third-order elastic,piezoelectric,and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials.It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups,while all third-order dielectric constants are 0 for 16 kinds of point groups as well as isotropic material.The reason is that some of the point groups are centrally symmetric,and the other point groups are high symmetry.These results provide the foundation to measure these constants,to choose material,and to research nonlinear problems.Moreover,these results are helpful not only for the study of nonlinear elastic and piezoelectric problems,but also for the research on flexoelectric effects and size effects.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11871073,11771016).
文摘Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of G.The least H-eigenvalue of A(G)refers to the least real eigenvalue of A(G)associated with a real eigenvector.In this paper,we obtain a perturbation result on the least H-eigenvalue of A(G)when a branch of G attached at one vertex is relocated to another vertex,and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.
基金the National Natural Science Foundation of China(No.11271221)the Specialized Research Fund for State Key Laboratories.
文摘In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids.We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.
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.
基金This work was supported in part by the National Natural Science Foundation of China (Grant No. 11101263).
文摘A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 11471077).
文摘The k-uniform s-hypertree G = (V, E) is an s-hypergraph, where 1 ≤ s ≤ k - 1, and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree A. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just (△S/k).
基金Science and Technology Foundation of Guizhou Province (Qian Ke He Ji Chu [2016]1161)Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2016]255)+9 种基金Doctoral Scientific Research Foundation of Zunyi Normal College (BS[2015]09)Yanmin Liu was supported by the National Natural Science Foundations of China (Grant No. 71461027)Science and Technology Talent Training Object of Guizhou Province Outstanding Youth (Qian Ke He Ren Zi [2015]06)Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2014]295), 2013. 2014, 2015 Zunyi 15851 Talents Elite Project FundingZunyi Innovative Talent Team (Zunyi KH (2015)38)Junkang Tian was supported by the Natural Science Foundation of Guizhou Province (Qian Jiao He KY [2015]451)Science and Technology Foundation of Guizhou Province (Qian Ke He J Zi [2015]2147)Xianghu Liu was supported by the Guizhou Province Department of Education Fund (KY[2015]391,[2016]046)Guizhou Province Department of Education Teaching Reform Project ([2015]337)Guizhou Province Science and Technology Fund (qian Ke He Ji Chu [2016]1160).
文摘Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.
基金Natural Science Foundation of China(Grant Nos.11871073,11871077)NSF of Department of Education of Anhui Province(Grant No.KJ2017A362)。
文摘The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph.It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph,and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph.In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius,where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.