Determining finite connected graphs along the quadratic embedding constants of paths

Abstract

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