On graphs whose Hosoya indices are primitive Pythagorean triples

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
Full paper (PDF, 206 Kb)

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

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

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3. Haruo Hosoya (2007) Mathematical meaning and importance of the topological index Z, Croatica Chemica Acta, 80(2), 239–749.
4. 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).
5. Haruo Hosoya (2009) Pythagorean triples. II. Growing caterpillar graphs generating Pythagorean triples, Natur. Sci. Rep. Ochanomizu Univ., 59(2), 15–25. MR 2515294
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