On graphs with just three distinct eigenvalues

Citation

Rowlinson P (2016) On graphs with just three distinct eigenvalues. Linear Algebra and Its Applications, 507, pp. 462-473. https://doi.org/10.1016/j.laa.2016.06.031

Abstract
Let G be a connected non-bipartite graph with exactly three distinct eigenvalues Rho, mu, lambda, where Rho >mu >lambda. In the case that G has just one non-main eigenvalue, we find necessary and sufficient spectral conditions on a vertex-deleted subgraph of G for G to be the cone over a strongly regular graph. Secondly, we determine the structure of G when just mu is non-main and the minimum degree of G is 1 + mu − lambda mu: such a graph is a cone over a strongly regular graph, or a graph derived from a symmetric 2-design, or a graph of one further type.

Keywords
Main eigenvalue; Minimum degree; Strongly regular graph; Symmetric 2-design; Vertex-deleted subgraph

Journal
Linear Algebra and Its Applications: Volume 507

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