# Spectral Graph Theory: Course Outline (MAT 624 )

In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.

## Course Outline

Graph Spectrum Theory:

Matrices associated to a graph, the spectrum of various graphs, diameter, regular graphs, spanning trees, complete graphs, complete bipertite graphs, Calay graphs.

Some Modules on the Applications of Spectral Graph Theory:

(Introduction to spectral geometry of graphs; Courant-Fischer theorem and graph colorings; Inequalities and bounds on eigenvalues; graph approximations; Cheeger's inequalities; Diffusion on graphs; Discretizations of heat kernels.

Energy of Graph:

Laplacian Energy, Signless Energy, Distance Energy, Normalized Laplacian Energy

He-matrix for HoneyComb Graph:

Honeycomb graph, He matrix, Spectral radius of He-matrix and bounds, He-matrix with Integer spectrum, number of triangles of He-matrix.

Energy of He-matrix for Hexagonal System:

Energy of He matrix, Upper bounds for the energy, Energy of Coalescence of graphs and various theorems.

Degree Sequence:

Properties of Degree Sequences, Degree Sequences of Edge colored Graphs.