共查询到20条相似文献,搜索用时 31 毫秒
1.
In a paper Fallat et al. (J Graph Theory 50 (2005), 162–174) consider the question of the existence of simple graphs on n vertices whose Laplacian matrix has an integral spectrum consisting of simple eigenvalues only in the range , 0 always being, automatically, one of the eigenvalues. They completely characterize the case when n is one of the eigenvalues, but for the case when n is not, they conjecture that there are no such graphs. In that paper it is shown that, indeed, there are no such graphs for . In this paper we show that the conjecture is true for We actually consider the nonexistence of graphs whose Laplacians are realized by more general spectra , with , , , , and , subject to certain trace conditions. We show that, indeed, for sufficiently large n such graphs do not exist. Our methods are both graph theoretical and algebraic. In certain cases we refine the Cauchy interlacing theorem. Finally, rather than work with Laplacians which have nonpositive off‐Diagonal entries, we transform the problems to the realizability of spectra of nonnegative matrices which we term anti‐Laplacians. 相似文献
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In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly. 相似文献
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Sivaramakrishnan Sivasubramanian 《Discrete Mathematics》2009,309(10):3458-3462
Brendan McKay gave the following formula relating the average distance between pairs of vertices in a tree T and the eigenvalues of its Laplacian:
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Laplacian matrices and their spectrum are of great importance in algebraic graph theory. There exist efficient formulations for eigensolutions of the Laplacian matrices associated with a special class of graphs called product graphs. In this paper, the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and/or members, is investigated. The eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model. Implicitly restarted Lanczos method is employed to obtain the required eigenpairs of the reduced problem. Examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the IRL method. 相似文献
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Perron components and algebraic connectivity for weighted graphs 总被引:8,自引:0,他引:8
The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedler's original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure. 相似文献
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M.A. Fiol 《Linear algebra and its applications》1999,290(1-3):275-301
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some “weights” (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights “regularize” the graph, and hence allow us to define a kind of regular partition, called “pseudo-regular,” intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lovász, it is shown that the weight Shannon capacity Θ* of a connected graph Γ, with n vertices and (adjacency matrix) eigenvalues λ1 > λ2 … λn, satisfies where Θ is the (standard) Shannon capacity and v is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. 相似文献
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We investigate how the algebraic connectivity of a weighted tree behaves when the tree is perturbed by removing one of its branches and replacing it with another. This leads to a number of results, for example the facts that replacing a branch in an unweighted tree by a star on the same number of vertices will not decrease the algebraic connectivity, while replacing a certain branch by a path on the same number of vertices will not increase the algebraic connectivity. We also discuss how the arrangement of the weights on the edges of a tree affects the algebraic connectivity, and we produce a lower bound on the algebraic connectivity of any unweighted graph in terms of the diameter and the number of vertices. Throughout, our techniques exploit a connection between the algebraic connectivity of a weighted tree and certain positive matrices associated with the tree. 相似文献
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This article describes the structure of the graph minimizing the algebraic connectivity among all connected graphs made with some given blocks with fixed number of pendant blocks, the blocks that have exactly one point of articulation. As an application, we conclude that over all graphs made with given blocks, the algebraic connectivity is minimum for a graph whose block structure is a path. 相似文献
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Let G be a connected graph of order n. The diameter of G is the maximum distance between any two vertices of G. In the paper, we will give some lower bounds for the algebraic connectivity and the Laplacian spectral radius of G in terms of the diameter of G. 相似文献
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Algebraic connectivity of directed graphs 总被引:1,自引:0,他引:1
Chai Wah Wu 《Linear and Multilinear Algebra》2005,53(3):203-223
We consider a generalization of Fiedler's notion of algebraic connectivity to directed graphs. We show that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs. We prove inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number. Finally, we illustrate an application to the synchronization in networks of coupled chaotic systems. 相似文献
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Alessandro Savo 《Annals of Global Analysis and Geometry》2009,35(1):39-62
We study the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute
or relative boundary conditions. We prove a series of lower bounds when the domain is starlike or p-convex and the ambient
manifold has pinched negative curvature. The bounds are sharp for starlike domains. We then compute the asymptotics of the
first eigenvalue of hyperbolic balls of large radius. Finally, we give lower bounds also for Euclidean domains.
相似文献
13.
Chai Wah Wu 《Linear and Multilinear Algebra》2013,61(3):203-223
We consider a generalization of Fiedler's notion of algebraic connectivity to directed graphs. We show that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs. We prove inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number. Finally, we illustrate an application to the synchronization in networks of coupled chaotic systems. 相似文献
14.
Fielder [M. Fielder, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305] has turned out that G is connected if and only if its algebraic connectivity a(G)>0. In 1998, Fallat and Kirkland [S.M. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74] posed a conjecture: if G is a connected graph on n vertices with girth g≥3, then a(G)≥a(Cn,g) and that equality holds if and only if G is isomorphic to Cn,g. In 2007, Guo [J.M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711] gave an affirmatively answer for the conjecture. In this paper, we determine the second and the third smallest algebraic connectivity among all unicyclic graphs with vertices. 相似文献
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In the following, we consider some cases where the point spectrum of an operator is either very stable or very unstable with respect to small perturbations of the operator. The main result is about the shift operator on whose point spectrum is what we will call strongly stable. We also give some general perturbation results, including a result about the size of the set of operators that have an eigenvalue.
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K.L. Patra 《Linear algebra and its applications》2008,428(4):855-864
Let G=(V,E) be a tree on n?2 vertices and let v∈V. Let L(G) be the Laplacian matrix of G and μ(G) be its algebraic connectivity. Let Gk,l, be the graph obtained from G by attaching two new paths P:vv1v2…vk and Q:vu1u2…ul of length k and l, respectively, at v. We prove that if l?k?1 then μ(Gk-1,l+1)?μ(Gk,l). Let (v1,v2) be an edge of G. Let be the tree obtained from G by deleting the edge (v1,v2) and identifying the vertices v1 and v2. Then we prove that As a corollary to the above results, we obtain the celebrated theorem on algebraic connectivity which states that among all trees on n vertices, the path has the smallest and the star has the largest algebraic connectivity. 相似文献
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A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra. 相似文献
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In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results. 相似文献
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In this paper we consider the Schwarz radical of linear algebraic semigroups as defined in semigroup theory. We give some new characterizations of the complete regularity, regularity and solvability of irreducible linear algebraic monoids in terms of Schwarz radical data. Moreover, we give a generalization about the results of the kernel to the results of completely regular \(\mathscr {J}\)-classes. 相似文献