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.
To get a decomposed form, use (download)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[d,N]
computes the d-ary F-sequence of N.
FestoonTreeSymbolic[d,n] gives a symbolic decomposition of the 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[b1,...,bn] denotes a rooted tree with branches 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[T,x] and the energy of a F-tree.
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.