By Thomas Puppe

Graph Drawing is the technological know-how of discovering an intuitive visualization of a community (or in mathematical phrases of a graph). One strategy is to outline strength services that characterize layout standards for graph layouts. It occurs to be that the eigenvalues of graph comparable matrices are in the neighborhood optimum suggestions for the various strength features. utilizing the eigenvalues for a graph structure is termed Spectral Graph Drawing.This publication is a survey of Spectral Graph Drawing tools. Graph layouts of a number of graph-related matrices, akin to the adjacency or the Laplace matrix, are studied. there's a detailed part at the implementation of the graph layouts utilizing the ability new release. on the finish the point of interest is prolonged to the precise necessities for Dynamic Spectral Graph Drawing, i.e. time-variant graphs are drawn with spectral equipment.

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CHAPTER 5. CHARACTERIZING SPECTRAL GRAPH LAYOUTS 60 These graphs have no loosly connected nodes, that could easily be pushed to the boundary. It is optimal for them just to minimize over the weighted edge lengths. We now examine the deviation of node i in the dimension k from being placed in the weighted centroid cω (i): (k) (k) xi − cω (i) = = = = (k) xi − (k) di xi j∈N (i) ωij xj di − ωii (k) − ωii xi − (k) j∈N (i) ωij xj di − ωii (k) + ρdi xi di − ωii (k) λLρ xi λLρ + ρdi di − ωii (k) xi . The deviation depends on the degree, on ρ and on the eigenvalue λLρ corresponding to the axis vector x(k) .

2). 13 Given is a generalized Laplace matrix LG with positive degrees. Then yields for all vectors x ∈ Rn , x = 0: x T LG x = ωij (i,j)∈E xi xj − di dj (xi − xj ) . CHAPTER 4. 14 Given is a graph G with positive degrees, its generalized Laplace matrix LG and its relaxed Laplace matrix Lρ . Then yields for all ρ ∈ R: u ∈ Rn is an eigenvector of LG with eigenvalue λ, iff u is an eigenvector of D−1 Lp with eigenvalue λ − ρ. Proof: The following expressions are equivalent: D−1 Lu Lu Lu − ρDu Lρ u −1 D Lρ u = = = = = λu λDu λDu − ρDu (λ − ρ)Du (λ − ρ)u ✷ CHAPTER 4.

E. if all degrees of G are equal. 9 Given is a graph G with Laplace matrix L and the relaxed Laplace matrix Lρ . Then following is equivalent: a) The matrices L and Lρ have the the same eigenvectors. b) The vector 1 is an eigenvector of Lρ . c) The graph G is regular or ρ = 0. Proof: a) ⇒ b) is clear. b) ⇒ c) holds, since .. Lρ 1 = di − ρdi − .. .. n j=1 ωij = −ρ di .. . c) ⇒ a) holds, because if ρ = 0, then L = Lρ . And if G regular, then Lρ = L − ρdI, d ∈ R the node degree of G.