Spectral clustering algorithms are often used to find clusters in the community detection problem. Recently, a degree-corrected spectral algorithm was proposed. However, it is only for partitioning graphs which generated from stochastic blockmodels. This paper studies based on graph theory and shows that gives good approximation of optimal wide class graphs. Moreover, we also give theoretical s...

For sufficiently large subsets A,B, C,D of Fq, Gyarmati and Sárközy (2008) showed the solvability of the equations a+b = cd and ab+1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. They asked whether one can extend these results to every k ∈ N in the following way: for large subsets A,B, C,D of Fq, there are a1, . . . , ak, a1, . . . , ak ∈ A, b1, . . . , bk, b ′ 1, . . . , b ′ k ∈ B with ai + bj , aibj ...

Many approaches to analyzing the structure of a musical recording involve detecting sequential patterns within a selfsimilarity matrix derived from time-series features. Such patterns ideally capture repeated sequences, which then form the building blocks of large-scale structure. In this work, techniques from spectral graph theory are applied to analyze repeated patterns in musical recordings....

Despite of the extreme success of the spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these ve...

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M -theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory ...

Spectral graph theory is undoubtedly the most favored graph data analysis technique, both in theory and practice. It has emerged as a versatile tool for a wide variety of applications including data mining, web search, quantum computing, computer vision, image segmentation, and among others. However, the way in which spectral graph theory is currently taught and practiced is rather mechanical, ...

The analysis of matrices associated with discrete, pairwise comparisons can be a very useful toolbox for a computer scientist. In particular, Spectral Graph Theory is based on the observation that eigenvalues and eigenvectors of these matrices betray a lot of properties of graphs associated with them. The first major section of this paper is a survey of key results in Spectral Graph Theory. The...

We often think of graphs geometrically, i.e. we see vertices as points on a plane or in space, with edges as lines connecting them. When speci ed algebraically (e.g. by its adjacency matrix), the geometrical imagery is absent. Can we somehow create it from the algebraic description? The geometric representation might be useful in itself, say because we want to draw the graph on paper. But once ...

In this second talk we will introduce the Rayleigh quotient and the CourantFischer Theorem and give some applications for the normalized Laplacian. Our applications will include structural characterizations of the graph, interlacing results for addition or removal of subgraphs, and interlacing for weak coverings. We also will introduce the idea of “weighted graphs”.