This paper develops the necessary tools to understand the relationship between eigenvalues of the Laplacian matrix of a graph and the connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for dregular graphs) which bounds the number of...

We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this graph are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding...

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN N, we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we pro...

Given a projective plane Σ and a polarity θ of Σ, the corresponding polarity graph is the graph whose vertices are the points of Σ, and two distinct points p1 and p2 are adjacent if p1 is incident to p θ 2 in Σ. A well-known example of a polarity graph is the Erdős-Rényi orthogonal polarity graph ERq, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an uppe...

For example, the pentagon is strongly regular with parameters (v, k, λ, μ) = (5, 2, 0, 1). One easily verifies that a graph G is strongly regular with parameters (v, k, λ, μ) if and only if its complement G is strongly regular with parameters (v, v−k− 1, v− 2k+μ− 2, v−2k+λ). The line graph of the complete graph of order m, known as the triangular graph T (m), is strongly regular with parameters...

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A (κ, τ)-regular set is a subset of the vertices inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbours in it. We consider the case when κ = τ...

Graph partitioning problems are NP-complete and various heuristic algorithms exist in the litterature. Particularly, spectral graph partitioning algorithms partition the graph using the eigenvector associated with the second smallest eigenvalue of the \graph Laplacian." Through the use of graph theory we have devoloped preconditioned subspace algorithms for spectral partitioning.