Lethbridge Number Theory and Combinatorics Seminar: Himanshu Gupta

This talk focuses on the inverse eigenvalue problem for graphs (IEPG), which seeks to determine the possible spectra of symmetric matrices associated with a given graph G. These matrices have off-diagonal non-zero entries corresponding to the edges of G, while diagonal entries are unrestricted. A key parameter in IEPG is q(G), the minimum number of distinct eigenvalues among such matrices. The Johnson and Hamming graphs are well-studied families of graphs with many interesting combinatorial and algebraic properties. We prove that every Johnson graph admits a signed adjacency matrix with exactly two distinct eigenvalues, establishing that its q-value is two. Additionally, we explore the behavior of q(G) for Hamming graphs. This is a joint work with Shaun Fallat, Allen Herman, and Johnna Parenteau.