a classification of graphs through quadratic embedding constants and clique graph insights

The quadratic embedding constant (qec) of a graph $g$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $mathrm{qec}(g)$. by observing graph structure of the maximal cliques (clique graph), we show that a graph $g$ with $mathrm{qec}(g)<-1/2$ admits a ``cactus-like’’ structure. we derive a formula for the quadratic embedding constant of a graph consisting of two maximal cliques. as an application we discuss characterization of graphs along the increasing sequence of $mathrm{qec}(p_d)$, where $p_d$ is the path on $d$ vertices. in particular, we determine graphs $g$ satisfying $mathrm{qec}(g)