Classification of Graphs Using Quadratic Embedding Constants

Let $G=(V,E)$ be a finite connected graph and $D=[d(x,y)]_{x,y\in V}$ its distance matrix. The quadratic embedding constant (QEC) of a graph $G$ is defined by the conditional maximum of $\langle f, Df\rangle$, $f\in C(V)$, subject to two constraints $\langle f,f\rangle=1$ and $\langle \mathbf{1},f\rangle=0$. The QEC, introduced by Obata--Zakiyyah (2018), has recently been the focus of research as a new invariant for classifying graphs. In this talk, recalling the fundamental properties obtained so far, we discuss some new achievements on characterization of graphs $G$ with $\mathrm{QEC}(G)<-1/2$ and propose some challenges. This talk is partially based on the joint work with W. Młotkowski (Wroclaw) and E.T. Baskoro (Bandung).