Clustering Based on Eigenvectors of the Adjacency Matrix

The paper presents a novel spectral algorithm EVSA (eigenvector structure analysis), which uses eigenvalues and eigenvectors of the adjacency matrix in order to discover clusters. Based on matrix perturbation theory and properties of graph spectra we show that the adjacency matrix can be more suitable for partitioning than other Laplacian matrices. The main problem concerning the use of the adjacency matrix is the selection of the appropriate eigenvectors. We thus propose an approach based on analysis of the adjacency matrix spectrum and eigenvector pairwise correlations. Formulated rules and heuristics allow choosing the right eigenvectors representing clusters, i.e., automatically establishing the number of groups. The algorithm requires only one parameter-the number of nearest neighbors. Unlike many other spectral methods, our solution does not need an additional clustering algorithm for final partitioning. We evaluate the proposed approach using real-world datasets of different sizes. Its performance is competitive to other both standard and new solutions, which require the number of clusters to be given as an input parameter.

An Interesting Property of a Class of Circulant Graphs

Suppose thatΠ=Cay(Zn,Ω)andΛ=Cay(Zn,Ψm)are two Cayley graphs on the cyclic additive groupZn, wherenis an even integer,m=n/2+1,Ω=t∈Zn∣t is odd, andΨm=Ω∪{n/2}are the inverse-closed subsets ofZn-0. In this paper, it is shown thatΠis a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum ofΠ. Finally, we show that ifn≥8andn/2is an even integer, then the adjacency matrix spectrum ofΛisn/2+11,1-n/21,1n-4/2,-1n/2(we write multiplicities as exponents).