Constructions for cospectral graphs for the normalized Laplacian matrix and distance matrix

In discrete mathematics, a graph is a representation of relationships between objects. Using linear algebraic techniques, we can encode a graph into a matrix. However, as the graph grows, so too does the matrix. This leads to computational limitations and necessitates the development of techniques to capture a portion of the graph\u27s structure. Spectral graph theory is one such method, which looks at the eigenvalues (or spectrum) of the matrix associated with the graph. We know only a portion of the structure is captured by the existence of cospectral graphs, or fundamentally different graphs with the same eigenvalues. Exploring cospectral graphs helps us understand which information is contained in the spectrum. In this thesis, we describe methods of creating cospectral graphs, specifically those with differing numbers of edges

Constructions for cospectral graphs for the normalized Laplacian matrix and distance matrix

In discrete mathematics, a graph is a representation of relationships between objects. Using linear algebraic techniques, we can encode a graph into a matrix. However, as the graph grows, so too does the matrix. This leads to computational limitations and necessitates the development of techniques to capture a portion of the graph's structure. Spectral graph theory is one such method, which looks at the eigenvalues (or spectrum) of the matrix associated with the graph. We know only a portion of the structure is captured by the existence of cospectral graphs, or fundamentally different graphs with the same eigenvalues. Exploring cospectral graphs helps us understand which information is contained in the spectrum. In this thesis, we describe methods of creating cospectral graphs, specifically those with differing numbers of edges.</p

Codeterminantal graphs

We introduce the concept of codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. To do this, we investigate the relationship of the spectrum and the Smith normal form (SNF) with the determinantal ideals. We establish a necessary and sufficient condition for graphs to be codeterminantal on R[x], and we present some computational results on codeterminantal graphs up to 9 vertices. Finally, we show that complete graphs and star graphs are determined by the SNF of its distance Laplacian matrix

On the Distance Spectra of Graphs

The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of double odd graphs and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.This is a manuscript of an article published as Aalipour, Ghodratollah, Aida Abiad, Zhanar Berikkyzy, Jay Cummings, Jessica De Silva, Wei Gao, Kristin Heysse et al. "On the distance spectra of graphs." Linear Algebra and its Applications 497 (2016): 66-87. DOI: 10.1016/j.laa.2016.02.018. Posted with permission.</p