vpar.gp

This is some functions for Pari/GP making calculations on labelled oriented trees and forests (graph theory trees) as "vpar" vector of parent vertex numbers. Various things treat as unlabelled and/or free too. Functions include

• Height, depth, eccentricity, radius, diameter, centre, centroid, weights, Wiener index and terminal Wiener.
• Children, neighbours, siblings, childless, leaves, singletons, rows, min/max ancestor, min/max descendant.
• Matrix adjacency, incidence, path lengths, reachability, Laplacian. Characteristic polynomial of adjacency, Laplacian.
• Independent sets, matchings, maximal matchings, equimatchable, stable equimatchable, almost equimatchable, edge covers.
• Dominating sets, minimal dominating sets, independent dominating sets, total dominating sets, semi-total domination number, perfect dominating sets, disjoint domination number.
• Relabelling, re-rooting, upwards/downwards.
• Isomorphism as unlabelled, free, ordered. Canonical unlabelled, free, ordered, unrooted, lexmin.
• Automorphisms, orbits, asymmetric, rooted or free.
• Iterate labelled, unlabelled, free, preorder. I'th preorder or downwards.
• Prime code (Matula-Goebel), balanced binary, graph6, sparse6, digraph6.
• Subdivide, join, delete, contract, branch reduce, root above, forest components.
• Sequences pre-order, post-order, rows, up ascend (Prüfer), up queue, down queue, subtree height, subtree sizes, blob, successive paths, childless paths.
• Make some specific trees, and random labelled, preorder, downwards.
• Counts of various trees, forests, or properties.
• Least child monk, hereditary least single, strongly monotone, equaldepths.

Vertices are numbered 1 to n. A tree is a vector "vpar" where vpar[v] is parent of v, or 0 if no parent. Such a representation is oriented in that there is a distinguished root (or roots for a forest) and labelled in that each vertex has a particular number. There are no other attributes.

        3
       ^ ^          tree, root=3
      /   \
     1     5        vpar=[3,1,0,1,3]
    ^ ^
   /   \
  2     4

The main use is to calculate or verify properties of specific trees of interest. Various functions like diameter are the same for any root or labelling so effectively act as on a "free" tree and the vertex numbers just define the structure. Most functions are linear in the number of vertices so can be used on large trees.

Connections to Pari/GP specifics include polynomials for roots (spectra), matrices for some linear algebra, Set()s for independent sets, permutations in relabelling and automorphisms, and then general compactness of GP for experimenting etc.

vpar.gp is free software (free as in freedom), published under the terms of the GNU General Public License (v3 or higher). Download version 22 here,

vpar.gp (756k, and sig), or compressed vpar.gp.gz (201k)
vpar-22.tar.gz (1290k, and sig)

Just vpar.gp is enough to run. Comments at the start give an introduction and overview, then each function has an addhelp(). The tar file includes some example scripts, self-tests, and work-in-progress extras (vpar-wip.gp). These extras mostly work but may change wildly. The sig files are Gnu PG ascii armoured signatures generated from my key.

The examples include bits of

• Binomial tree Wiener index.
• Bulgarian solitaire.
• Count preorder forests by footprint (canopy) exercise from Knuth.
• Complete binary tree matchings.
• Cospectrals per Mowshowitz.
• Equal01 towards balanced by several authors, realizing the Chung-Feller theorem.
• Equimatchable tree counts and generating functions.
• Huffman code 1 to n tree.
• Integral trees (integer adjacency spectrum) by diameter per Csikvari.
• Isomorphic halves trees.
• Maximal asymmetric subtrees per Nesetril.
• Mean number of singletons per forest (rooted unlabelled).
• Most maximum matchings per Heuberger and Wagner.
• Parking function to labelled forest per de Oliveira, and las Vergnas.
• Path lengths determinants per Graham and Pollak, and Bapat and Sivasubramanian.
• Perm to upwards per Dennis Walsh.
• Random decorated binary tree per Jean-Luc Rémy.

Bugs

vpar_to_free() in version 19 and earlier had a bug where it did not give the intended free-primary form (per vpar_free_next() iteration). See note in vpar.gp for some details. The wrong form was still canonical (fine for unduplicating up to isomorphism etc), but was not the free-primary canonical form.