Today and next time, we will cover what is known as spectral graph partitioning, and in particular we will discuss and prove Cheeger’s Inequality. This result is central to all of spectral graph theory as well as a wide range of other related spectral graph methods. (For example, the isoperimetric “capacity control” that it provides underlies a lot of classification, etc. methods in machine lea...

The main topic of this course has been the unexpected connection between graph theory and spectral theory: we’ve seen the relationships between graph spectra and connectivity, bipartiteness, independent sets, colorings, trees, cuts, flows, and others. Today we see another connection with a standard concept in graph theory: planarity. In many branches of mathematics, it is frequently helpful to ...

Extended Abstract 1 Preliminaries. Following the spectral graph theory, a graph coloring technique and algebraic methods in graph theory (see [5], [13]), we continue a Coxeter spectral study the category Bigrn of connected non-negative loop-free edge-bipartite (signed) graphs ∆, with n ≥ 1 vertices (bigraphs, in short), and their morsifications introduced by the second named author in [22]-[25]...

Cámara and Haemers (2014) investigatedwhen a complete graphwith some edges deleted is determined by its adjacency spectrum (DAS for short). They claimed: for anym ≥ 6 and every large enough n one can obtain graphswhich are not DAS by removingm edges from a complete graph Kn. LetGn denote the set of all graphs obtained from a complete graph Kn by deleting six edges. In this paper, we show that a...

In this paper, we study explicit correspondences between the integrable Novikov and Sawada–Kotera hierarchies, and between the Degasperis–Procesi and Kaup–Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada–Kotera equations, and the isospectral problems of the Degasperis–Procesi and Kaup–Kupershmidt equations relate...

A version of Henrici's classical perturbation theorem for eigenvalues of matrices is obtained for joint spectra of commuting tuples of matrices. The approach involves Clifford algebra techniques introduced by Mcintosh and Pryde.

In this paper we construct, for n ≥ 2, arbitrarily large families of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada’s method.