Tomoe Kadoi and Yasuhide Numata

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 1, Pages 59—80

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## Details

### Authors and affiliations

Tomoe Kadoi

*Department of Mathematical Sciences
Shinshu University
3-1-1 Asahi, Matsumoto-shi, Nagano-ken, 390-8621, Japan
*

Yasuhide Numata

*Department of Mathematical Sciences*

Shinshu University

3-1-1 Asahi, Matsumoto-shi, Nagano-ken, 390-8621, Japan

Shinshu University

3-1-1 Asahi, Matsumoto-shi, Nagano-ken, 390-8621, Japan

### Abstract

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. In this paper, we show the uniqueness of the common structure.

### Keywords

- Caterpillar trees
- continuants
*Z*-indices- reduced Pythagorean triples
- continued fractions

### AMS Classification

- 05C30
- 05C70
- 05C05
- 05C50
- 11A05
- 11A55

### References

- Haruo Hosoya (1971) Topological index. A newly proposed quantity characterizing the

topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn, 44(9), 2332–2339 - Haruo Hosoya (2007) Continuant, caterpillar, and topological index
*Z*. Fastest algorithm for degrading a continued fraction, Natur. Sci. Rep. Ochanomizu Univ., 58(1), 15–28. MR2416190(2009e:11026) - Haruo Hosoya (2007) Mathematical meaning and importance of the topological index
*Z*, Croatica Chemica Acta, 80(2), 239–749. - Haruo Hosoya (2008) Pell equation. V. Systematic relation between the Pythagorean triples and Pell equations, Natur. Sci. Rep. Ochanomizu Univ., 59(1), 19–34. MR 2513595(2010e:11019).
- Haruo Hosoya (2009) Pythagorean triples. II. Growing caterpillar graphs generating Pythagorean triples, Natur. Sci. Rep. Ochanomizu Univ., 59(2), 15–25. MR 2515294
- Haruo Hosoya (2009) Pythagorean triples. III. Systematic generation of primitive Pythagorean triples by the topological index and caterpillar graphs, Natur. Sci. Rep. Ochanomizu Univ., 60(1), 15–30. MR 2732476

## Related papers

## Cite this paper

APAKadoi, T. & Numata Y. (2016). On graphs whose Hosoya indices are primitive Pythagorean triples. Notes on Number Theory and Discrete Mathematics, 22(1), 59-80.

ChicagoKadoi, Tomoe, and Yasuhide Numata. “On graphs whose Hosoya indices are primitive Pythagorean triples.” Notes on Number Theory and Discrete Mathematics 22, no. 1 (2016): 59-80.

MLAKadoi, Tomoe, and Yasuhide Numata. “On Graphs whose Hosoya Indices are Primitive Pythagorean Triples.” Notes on Number Theory and Discrete Mathematics 22.1 (2016): 59-80. Print.