Lower Bounds on the Odds Against Tree Spectral Sets

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive lengths is tree spectral if it is the path spectrum of a tree. We show that for each even integer s ⩾ 2 at least 34.57% of all subsets of the set { 2 , 3 , … , s } are tree spectral, and for each odd integer s ⩾ 2 at least 27.44% of all subsets of the set { 2 , 3 , … , s } are tree spectral.