Abstract
We give a survey of graph spectral techniques used in computer sciences. The survey consists of a description of particular topics from the theory of graph spectra independently of the areas of Computer science in which they are used. We have described the applications of some important graph eigenvalues (spectral radius, algebraic connectivity, the least eigenvalue etc.), eigenvectors (principal eigenvector, Fiedler eigenvector and other), spectral reconstruction problems, spectra of random graphs, Hoffman polynomial, integral graphs etc. However, for each described spectral technique we indicate the fields in which it is used (e.g. in modelling and searching Internet, in computer vision, pattern recognition, data mining, multiprocessor systems, statistical databases, and in several other areas). We present some novel mathematical results (related to clustering and the Hoffman polynomial) as well.
Highlights
In this paper we shall give a survey of parts of the theory of graph spectra which are useful in computer sciences
Graphs that are treated in computer sciences using graph spectra typically represent either some physical networks or data structures In the first case the graphs usually have a great number of vertices and they are called complex networks while in the second case graphs are of small dimensions
From the presented material one can see that a great part of the theory of graph spectra is really used in computer sciences
Summary
In this paper we shall give a survey of parts of the theory of graph spectra which are useful in computer sciences. Methods of data mining (in particular, spectral graph clustering) appear in computer vision, social networks and Internet search while several problems of combinatorial optimization are relevant for data mining (e.g., in clustering). Graphs that are treated in computer sciences using graph spectra typically represent either some physical networks (computer network, Internet, biological network, etc.) or data structures (documents in a database, indexing structure, etc.) In the first case the graphs usually have a great number of vertices (thousands or millions) and they are called complex networks while in the second case graphs are of small dimensions.
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