Title: Structural Approaches to Multiplicities of Eigenvalues in Graphs

Abstract: This talk focuses on the multiplicities of eigenvalue of symmetric matrices associated with simple graphs, where nonzero off-diagonal entries correspond to edge weights and zeros indicate non-edges. No restrictions are imposed on the diagonal entries. The interplay between graph structure and matrix spectrum provides rich combinatorial and algebraic insights.  In particular, we show how certain structural features, especially in trees, can determine or constrain some of the multiplicities of eigenvalues, as well as geometric relations between the eigenvalues. This helps in completely understanding some of the achievable multiplicity lists and all possible spectra with those multiplicities. This is joint work with Fallat, Hall, Levene, Meyer, Oblak and Smigoc.