The problem of partitioning a graph such that the number of edges incident to vertices in diierent partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing ll-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smal...

Spectral graph theory discusses about the algebraic properties of graphs based on spectrum a graph. This article investigated prism The method used in this research is circulant matrix. results showed that P2,s regular degree 3, for s odd and ≥ circulantt with spectrum.

In this lecture, we discuss some spectral techniques and their applications. The goal of spectral techniques is to study and deduce the characteristics of a matrix by looking at its spectrum (i.e. eigenvalues of matrix). Although spectral methods can be applied to any problem instance with matrix representation, we will be focussing on graph theoretic problems in this lecture. Over the past two...

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by ...

We develop (within a possibly new) framework spectral analysis and operator theory on (almost) general graphs and use it to study spectral properies of the graph-Laplacian and so-called graph-Dirac-operators. That is, we introduce a Hilbert space structure, being in our framework the direct sum of a node-Hilbert-space and a bond-Hilbert-space, a Dirac operator intertwining these components, and...

This is a collection of references for a series of lectures that I gave in the “boot camp” of the semester on spectral graph theory at the Simons Institute in August, 2014. To keep this document focused, I only refer to results related to the lectures in question, which means that my own work is disproportionally represented. I plan to post a better version of this document in the future, so pl...

Planar graphs originated with the studies of polytopes and of maps. The skeleton (edges) of a threedimensional polytope provide a planar graph. We obtain a planar graph from a map by representing countries by vertices, and placing edges between countries that touch each other. Assuming each country is contiguous, this gives a planar graph. While planar graphs were introduced for practical reaso...