Determining finite connected graphs along the quadratic embedding constants of paths

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is definition the maximum quadratic function associated to distance matrix on certain sphere codimension two. We prove that constants paths $P_n$ form strictly increasing sequence converging $-1/2$. Then we formulate problem determining all graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G)<\mathrm{QEC}(P_{n+1})$. answer given for $n=2$ and $n=3$ exploiting forbidden subgraphs $\mathrm{QEC}(G)<-1/2$ explicit star products complete graphs.