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For a tree  T ,denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is  T . It is known that. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and  k -linear if there are  k  of these high degree vertices.is known for generalized stars (1-linear trees), andis determined here for 2-linear trees. Three (and higher) linear trees offer considerable additional complications.

The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a 2-linear tree