Eigenspaces of Graphs

Abstract

Current research on the spectral theory of finite graphs may be seen as part of a wider effort to forge closer links between algebra and combinatorics (in particular between linear algebra and graph theory).This book describes how this topic can be strengthened by exploiting properties of the eigenspaces of adjacency matrices associated with a graph. The extension of spectral techniques proceeds at three levels: using eigenvectors associated with an arbitrary labelling of graph vertices, using geometrical invariants of eigenspaces such as graph angles and main angles, and introducing certain kinds of canonical eigenvectors by means of star partitions and star bases. One objective is to describe graphs by algebraic means as far as possible, and the book discusses the Ulam reconstruction conjecture and the graph isomorphism problem in this context. Further problems of graph reconstruction and identification are used to illustrate the importance of graph angles and star partitions in relation to graph structure. Specialists in graph theory will welcome this treatment of important new research.

Dynamic state determination of a software-defined network via dual basis representation

“…This translation and rotation are also demonstrated as the other two nodes are removed from the network. The state of each node can be described in terms of a magnitude and angle about the center of the basis [16]. It is more difficult to visualize these rotations in multi-dimensional space, but these rotations can be computed in the multidimensional space that is described by the dual basis.…”

The Local Spectra of Line Graphs

“…For a number of applications of these parameters, see for instance Cvetković, Rowlinson, and Simić [11].…”

Graph-Theoretical Methods in Computer Vision

“…In this case, the model DAG is a subgraph of the query DAG, or vice versa. The following theorem relates the eigenvalues of two such DAGs: Theorem 1 (see Cvetković et al [6] …”

“…6 Next, we form a bipartite edge weighted graph G (V 1 , V 2 , E G ) with edge weights from the matrix Π (G, H). 7 Using the scaling algorithm of Goemans, Gabow, and Williamson [11], we then find the maximum cardinality, minimum weight matching in G. This results in a list of node correspondences between G and H, called M 1 , that can be ranked in decreasing order of similarity.…”

“…6 2.9 0.5 trate the matching of an occluded image to the database, we compare an image containing the piggy bank occluded by the bulb socket, as shown in Fig. 10.…”