On graphs whose Hosoya indices are primitive Pythagorean triples
  We discuss families of triples of graphs whose Hosoya indices are primitive Pythagorean triples. Hosoya gave a method to construct such families of caterpillars, i.e., trees whose vertices are within distance 1 of a central path. He also pointed out a common structure to the families, and conjectured the uniqueness of the structure. In this paper, we give an answer to his conjecture. 
Tomoe Kadoi, Yasuhide Numata
math.CO