B. Mohar. The laplacian spectrum of graphs. In Sixth International Conference on the Theory and Applications of Graphs, pages 871--898, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....have been a number of attempts to apply spectral graph theory to problems in combinatorial optimization. For example, spectral algorithms have been developed for graph coloring [3] graph partitioning [9, 28] and envelope reduction [4] and more examples can be found in the survey papers of Mohar [23, 24]. However, in most previous applications, these techniques have been used to provide bounds, heuristics, or in a few cases, approximation algorithms [2, 6, 14] for NP hard problems. There are only a small number of previous results in which eigenvector techniques have been used to exactly solve ....

B. Mohar, The Laplacian spectrum of graphs, in Graph Theory, Combinatorics and Applications, Y. Alavi, G. Chartrand, O. Oellermann, and A. Schwenk, eds., Wiley, New York, 1991, pp. 871--898.

....classes of graphs and for obtaining bounds on properties such as the diameter, girth, chromatic number, connectivity, etc. 4, 13, 14, 34, 36] More recently, the interest has shifted somewhat from the adjacency spectrum to the spectrum of the closely related graph Laplacian, see e.g. [12, 38, 49, 50]. Again, the dominating part of the theory is concerned with the eigenvalues. The eigenvectors of graphs, however, have received only sporadic attention on their own. Even the recent book on Eigenspaces of Graphs [15] contains only a few pages on the geometric properties of the eigenvectors which ....

B. Mohar. The laplacian spectrum of graphs. In Y. Alavi, G. Chartrand, O. Ollermann, and A. Schwenk, editors, Graph Theory, Combinatorics, and Applications, pages 871--898, New York, 1991. John Wiley and Sons, Inc.

....system evolves. The model under consideration turns out to provide a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. In Section 2. 2 we define the notion of an average heading vector in terms of graph Laplacians [21] and we show how this idea leads naturally to the Vicsek model as well as to other decentralized control models which might be used for the same purposes. We propose one such model which assumes each agent knows an upper bound on the number of agents in the group, and we explain why this model has ....

....open loop control u. To end up with the Vicsek model, u would have to be defined as D #(t) 1 e(t) 21) where e is the average heading error vector e(t) L #(t) #(t) 22) and, for each p L p is the symmetric matrix L p = D p p (23) 9 known in graph theory as the Laplacian of G p [21, 25]. It is easily verified that equations (20) to (23) do indeed define the Vicsek model. We ve elected to call e the average heading error because if e(t) 0 at some time t, then the heading of each agent with neighbors at that time will equal the average of the headings of its neighbors. In the ....

B. Mohar. The Laplacian spectrum of graphs. in Graph theory, combinatorics and applications (Ed. Y. Alavi G. Chartrand, O. R. Ollerman, and A. J. Schwenk), 2:871--898, 1991.

....system evolves. The model under consideration turns out to provide a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. In Section 2. 2 we define the notion of an average heading vector in terms of graph Laplacians [27] and we show how this idea leads naturally to the Vicsek model as well as to other decentralized control models which might be used for the same purposes. We propose one such model which assumes each agent knows an upper bound on the number of agents in the group, and we explain why this model has ....

....open loop control u. To end up with the Vicsek model, u would have to be defined as D #(t) 1 e(t) 21) where e is the average heading error vector e(t) L #(t) #(t) 22) and, for each p L p is the symmetric matrix L p = D p p (23) known in graph theory as the Laplacian of G p [27, 31]. It is easily verified that equations (20) to (23) do indeed define the Vicsek model. We ve elected to call e the average heading error because if e(t) 0 at some time t, then the heading of each agent with neighbors at that time will equal the average of the headings of its neighbors. In the ....

B. Mohar. The Laplacian spectrum of graphs. in Graph theory, combinatorics and applications (Ed. Y. Alavi G. Chartrand, O. R. Ollerman, and A. J. Schwenk), 2:871--898, 1991.

....system evolves. The model under consideration turns out to provide a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. In Section 2. 2 we define the notion of an average heading vector in terms of graph Laplacians [21] and we show how this idea leads naturally to the Vicsek model as well as to other decentralized control models which might be used for the same purposes. We propose one such model which assumes each agent knows an upper bound on the number of agents in the group, and we explain why this model has ....

B. Mohar. The Laplacian spectrum of graphs. in Graph theory, combinatorics and applications (Ed. Y. Alavi G. Chartrand, O. R. Ollerman, and A. J. Schwenk), 2:871-- 898, 1991.

.... Hall [9] has shown that v is the best non trivial minimum of the following quadratic energy: E = v i v j ) 1) subject to: #v# 2 =1 As a matter of fact, v is a vector of fundamental importance to many fields as it reflects connectivity properties of the graph, see, e.g. [16]. So fundamental to be privileged in having a unique name the Fiedler vector. Sequencing the vertices is done by sorting them according to their components in v. Fortunately, computation of v can be done very rapidly using the multi scale (or multi level ) methods of [4] 14] In fact, in ....

B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

....corresponding real orthonormal eigenvectors by v 1 = 1 # n) 1 n ,v 2 , v n . To avoid v 1 , Hall s strategy is to take the next eigenvector, v 2 , as the vector of coordinates. As a matter of fact, v 2 is a vector of fundamental importance to many fields besides graph drawing; see, e.g. [21]. In fact, it has its own name the Fiedler vector. Relying on the orthogonality of the eigenvectors, v 2 is the minimizer of (3) in the subspace orthogonal to v 1 . Therefore, Hall s strategy can be formulated as: x =1 1 n =0. Appropriately, we henceforth term this the ....

B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

.... that v is the best non trivial minimum of the following quadratic energy: E = v i v j ) 1) subject to: #v# 2 =1 In fact, the vector v called the Fiedler vector is of fundamental importance in many fields, as it reflects connectivity properties of the graph; see, e.g. [18]. Sequencing the vertices is done by sorting them according to their components in v. Fortunately, computation of v can be carried out very rapidly using the multi scale (or multi level ) methods of [4, 16] 5 Function MinLA DP (G(V = Input: A graph G(V,E) Output: vector Ordering ....

B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

....real orthonormal eigenvectors by v 1 = 1 # n) 1 n , v 2 , v n . To avoid v 1 , Hall s strategy is to take the next eigenvector, v 2 , as the vector of coordinates. As a matter of fact, v 2 is a vector of fundamental importance to many fields besides graph drawing, see, e.g. [17], so fundamental to be privileged in having a unique name the Actually, for connected graphs L has one and only one zero eigenvalue, see Lemma B.1 Fiedler vector. Relying on the orthogonality of the eigenvectors, v 2 is the minimizer of (3) in the subspace orthogonal to v 1 . Therefore, ....

....A special case of a connected PSD graph with two zero eigenvalues is shown in Figure 12. The Laplacian is One can even extend this proof to show that the number of zero eigenvalues of the Laplacian of an AP graph G equals to the number of disconnected subgraphs composing it, see e.g. [17]. 32 and it has two zero eigenvalues corresponding to the orthogonal vectors v 1 = 1, 1, 1) v 2 = 1, 0, 1) Nonetheless, we can still say the following: Lemma B.2 Let G(L, M) be a PSD graph. If its Laplacian L has only one zero eigenvalue, then G is connected. Proof Similarly to the ....

B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

....Spectral methods have become standard techniques in algebraic graph theory. The most widely used techniques utilize eigenvalues and eigenvectors of the adjacency matrix of the graph. More recently, the interest has shifted somewhat to the spectrum of the closely related Laplacian. In fact, Mohar [9] claims that the Laplacian spectrum is more fundamental than this of the adjacency matrix. Related areas where the spectral approach has been popularized include clustering [12] partitioning [11] and ordering [6] However, these areas use discrete quantizations of the eigenvectors, unlike graph ....

....i i, j =1, n. We will often omit the G in A . The Laplacian is another symmetric n n matrix associated with the graph, denoted by L L deg(i) i = j w ij i i, j =1, n. Again, we will often omit the G in L . The Laplacian has many interesting properties (see, e.g. [9]) Here we state some useful features: L is a real symmetric and hence its n eigenvalues are real and its eigenvectors are orthogonal. L is positive semi definite and hence all eigenvalues of L are non negative. 1 n = 1, 1, 1) is an eigenvector of L, with the associated ....

B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

....of Fourier transform. A natural approach to adapt the construction of smoothing functional to functions de ned on a graph is therefore to adapt the Fourier transform to that context. 9 As a matter of fact Fourier transforms on graphs have been extensively studied in spectral graph theory [Chu97, Moh91, Moh97, Sta96] as we now recall. Let D be the n n diagonal matrix of vertex degrees of the graph , i.e. D x;y = 0 if x 6= y; deg(x) if x = y; where deg(x) is the number of edges involving x in , and let A be the adjacency matrix de ned by: A x;y = 1 if there is an edge ....

B. Mohar. The laplacian spectrum of graphs. In Y. Alavi, G. Chartrand, O. Ollermann, and A. Schwenk, editors, Graph theory, combinatorics, and applications, pages 871-898, New-York, 1991. John Wiley and Sons, Inc.

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B. Mohar, The Laplacian spectrum of graphs, in: "Graph Theory, Combinatorics, and Applications," (Y. Alavi et al., eds.), J. Wiley, New York, 1991, pp. 871--898.

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B. Mohar. The laplacian spectrum of graphs. In Sixth International Conference on the Theory and Applications of Graphs, pages 871--898, 1988.

No context found.

B. Mohar, "The laplacian spectrum of graphs," Graph Theory, Combinatorics, and Applications, Y. Alavi et al. (eds.), John Wiley, New York, NY, pp. 871-898, 1988.

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B. Mohar, "The Laplacian Spectrum of Graphs," Proc. Sixth Quadrennial Conf. Theory and Applications of Graphs, vol. 2, pp. 871898, 1998.

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B. Mohar. The Laplacian spectrum of graphs. In Sixth International Conference on the Theory and Applications of Graphs, pages 871--898, 1988.

No context found.

B. Mohar, \The Laplacian spectrum of graphs", Graph Theory, Combinatorics, and Applications, 2, 871-898, 1991.

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B. Mohar. The Laplacian spectrum of graphs. In Y. Alavi, G. Chartrand, and A. S. O.R. Oellermann, editors, Graph Theory, Combinatorics, and Applications, volume 2, pages 871--898. Wiley, 1991.

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B. Mohar. The laplacian spectrum of graphs. In Y. Alavi, editor, Graph Theory, Combinatorics and Applications, pages 871--898. J. Wiley, New York, 1991.

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B. Mohar. The Laplacian Spectrum of Graphs. Technical Report, Dept. of Mathematics, Univ. of Ljubljana, 61111 Ljubljana, Yugoslavia, 1988.

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B. Mohar, "The Laplacian Spectrum of Graphs", Graph Theory, Combinatorics, and Applications 2 (1991), 871--898.

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B. Mohar, "The Laplacian spectrum of graphs", Graph Theory, Combinatorics, and Applications, 2, 871--898, 1991.

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B. Mohar. The Laplacian spectrum of graphs. in Graph theory, combinatorics and applications (Ed. Y. Alavi G. Chartrand, O. R. Ollerman, and A. J. Schwenk), 2:871-898, 1991.

No context found.

B. Mohar. The laplacian spectrum of graphs. In Y. Alavi, G. Chartrand, O. Ollermann, and A. Schwenk, editors, Graph Theory, Combinatorics, and Applications, pages 871--898, New York, 1991. John Wiley and Sons, Inc.

No context found.

Bojan Mohar. The Laplacian spectrum of graphs. In Graph theory, combinatorics, and applications. Vol. 2 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., pages 871-898. Wiley, New York, 1991.

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