Determining finite connected graphs along the quadratic embedding constants of paths - Edy Tri Baskoro; Nobuaki Obata (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2021)
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.