Spectral graph theory studies the structural properties of graphs by associating various types of matrices with them. The spectrum of the adjacency matrix of a graph G reveals the number of edges and bipartiteness of G and provides bounds for the chromatic number and independence number of G. The Laplacian spectrum of G reveals the number of connected components and spanning trees of G, and it also provides bounds for the connectivity of G. However, not all features are revealed by the spectrum of the associated matrices. One such instance is that two nonisomorphic graphs can have the same spectrum. In this talk, we shall first recall the basic notions of spectral graph theory. Then, we discuss some of the graphs which are determined by their spectrum and constructions of cospectral nonisomorphic graphs. Part of this talk is a joint work with Shivaramakrishna Pragada and Hitesh Wankhede.
