This page contains online ressources for the paper “On a Class of Extremal Trees for Various Indices” by C. Heuberger and S. Wagner.

In festoontree.asy, we provide an Asymptote package for drawing F-trees.

Currently (as of asymptote 1.43), a patched copy of drawtree.asy is required.

**Usage:** Copy the two files into one directory. A sample usage would be (download)

import festoontree; FestoonTree(3,69);

This creates the F-tree for `d`=3 and `n`=69 in explicit form.

import festoontree; FestoonTree(3,69,true);

Running `asy`

on these files yields .eps files containing the graphs.

For up to 100 vertices and `d` at most 5, we have compiled pdf files containing the F-trees.

d |
Decomposed Form | Explicit Form |
---|---|---|

2 | 305 KB | 162 KB |

3 | 311 KB | 151 KB |

4 | 316 KB | 153 KB |

5 | 320 KB | 149 KB |

The file FestoonTree.m contains Mathematica^{®} code for creating F-trees and
computing their Merrifield-Simmons and Hosoya indices as well as their energy.

An examples file is also available.

<<FestoonTree.m

loads the package

FestoonSequence[3,139] FestoonTreeSymbolic[3,69] FestoonTreeSemiSymbolic[3,69] FestoonTree[3,69] FestoonTree[3,69]//TreeForm

Computes the F-sequence as well as the F-trees. `FestoonSequence[`

computes the `d`,`N`]`d`-ary F-sequence of `N`.
`FestoonTreeSymbolic[`

gives a symbolic decomposition of the `d`,`n`]`d` ary F-tree of order `n`,
in `FestoonTreeSemiSymbolic`

the Festoon is already replaced by Tree constructs, and in `FestoonTree`

, the complete `d`-ary trees are also translated into Tree constructs.

`Tree[`

denotes a rooted tree with branches `b1`,...,`bn`]`b1`, ..., `bn`.

sigma[FestoonTree[3,69]] z[FestoonTree[3,69]] M[FestoonTree[3,69],x] Energy[FestoonTree[3,69]] N[Energy[FestoonTree[3,69]]]

Computing the Merrifield-Simmons index `sigma`

, the Hosoya index `z`

, the matching polynomial `M[`

and the energy of a F-tree.`T`,`x`]

sigma[FestoonTreeSemiSymbolic[3,69]] z[FestoonTreeSemiSymbolic[3,69]] M[FestoonTreeSemiSymbolic[3,69],x] Energy[FestoonTreeSemiSymbolic[3,69]] N[Energy[FestoonTreeSemiSymbolic[3,69]]]

Working with `FestoonTreeSemiSymbolic`

(or `FestoonTreeSymbolic`

) increases the performance of the index calculations,
as intermediate results for complete trees are cached.

C. Heuberger, S. Wagner. Last Modification: 2008-08-27