Using the technique of ”semantic mirroring” a graph is obtained that represents words and their translations from a parallel corpus or a bilingual lexicon. The connectedness of the graph holds information about the different meanings of words that occur in the translations. Spectral graph theory is used to partition the graph, which leads to a grouping of the words according to different senses...

In this thesis, we aim to use the spectral graph theory to develop a framework to solve the problems of computer vision. The graph spectral methods are concerned with using the eigenvalues and eigenvectors of the adjacency matrix or closely related Laplacian matrix. In this thesis we develop four methods using spectral techniques: (1) We use a Hermitian property matrix for point pattern matchin...

We study a family of graphs related to the n-cube. The middle cube graph of parameter k is the subgraph of Q2k−1 induced by the set of vertices whose binary representation has either k − 1 or k number of ones. The middle cube graphs can be obtained from the wellknown odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theor...

Parallel Repetition governs the decay of the verification probability in a repeated 2-prover interactive protocol. We will first analyze Feige’s game which shows the naive exponential bound does not hold. Then we will state Raz’s original bound[6] and observe some interesting properties of it. Though the original proof by Raz is very involved, we will give a high level proof idea and analyze ma...

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors ar...

This survey accompanies the Josiah Williard Gibbs Lecture that I gave at the 2016 Joint Mathematics Meetings. It provides an introduction to three topics: algebraic and spectral graph theory, the sparsification of graphs, and the recent resolution of the Kadison–Singer Problem. The first and third are connected by the second.

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenva...

Let G be a connected graph on n vertices, and let D(G) be the distance matrix of G. Let ∂1(G) ≥ ∂2(G) ≥ · · · ≥ ∂n(G) denote the eigenvalues of D(G). In this paper, the connected graphs with ∂n−1(G) at least the smallest root of x3 − 3x2 − 11x− 6 = 0 are determined. Additionally, some non-isomorphic distance cospectral graphs are given.