Algebraic Graph Theory-Shengtong Zhang

Title: Squares of eigenvalues and semi-definite optimization

Abstract: I will share some recent progress on two long-standing conjectures in spectral graph theory, namely the Elphick-Farber-Goldberg-Wocjan conjecture and the Bollob\'{a}s-Nikiforov conjecture. Both conjectures involve bounds on the sum of squares of the eigenvalues of a graph, and a key ingredient in our work is the interpretation of such sums as optimization problems involving semi-definite matrices. Part of the talk is joint work with Gabriel Coutinho and Thomás Jung Spier.