Minimum laziness among graphs

Determine whether the path P_n is among the graphs on n vertices that minimize the Laplacian laziness tr(X).

Background

The paper establishes several upper-extremal results for graph laziness but does not characterize the graphs attaining the minimum. The authors report that data on graphs with up to nine vertices raises the question of whether the path P_n belongs to the minimizing family among all graphs on n vertices.

References

Is Pn one of the graph that minimize tr(X) among all graphs on nvertices?

— Laziness of Quantum Walks on Graphs  (2608.20739 - Mohan et al., 21 Aug 2026) in Section 8, “Future work,” item (ii), p. 46

It is conjectured in [13] that Kn maximizes tr(X) relative to the adjacency matrix over all connected graphs on nvertices.

— Laziness of Quantum Walks on Graphs  (2608.20739 - Mohan et al., 21 Aug 2026) in Section 8, “Future work,” p. 47