Classify graphs whose quadratic embedding constant equals the second distance-matrix eigenvalue

Determine the class of finite connected graphs G for which the quadratic embedding constant QEC(G) equals the second largest eigenvalue of the distance matrix of G.

Background

For a connected graph G, the quadratic embedding constant is defined by maximizing the quadratic form associated with the distance matrix over unit vectors orthogonal to the all-one vector. For transmission-regular graphs, the all-one vector spans the eigenspace of the largest distance-matrix eigenvalue, and the variational characterization therefore gives QEC(G) equal to the second largest distance-matrix eigenvalue.

The paper notes that the equality is known for every transmission-regular graph and for additional examples, including paths with an even number of vertices, but does not characterize the full class of graphs satisfying it. The open problem is consequently to determine exactly which connected graphs have this equality.