- HOUSE OF GRAPHS
- SEE ALSO
Graph::Maker::BinomialTree - create binomial tree graph
use Graph::Maker::BinomialTree; $graph = Graph::Maker->new ('binomial_tree', N => 32);
Graph::Maker::BinomialTree creates a
Graph.pm graph of a binomial tree with N vertices. Vertices are numbered from 0 at the root through to N-1.
The parent of vertex n is that n with its lowest 1-bit cleared to 0. Conversely the children of a vertex n=xx1000 are xx1001, xx1010, xx1100, so each trailing 0 bit changed to a 1, provided that does not exceed the maximum N-1. At the root the children are single bit powers-of-2 up to the high bit of the maximum N-1.
__0___ / | \ N => 8 1 2 4 | / \ 3 5 6 | 7
order parameter is another way to specify the number of vertices. A tree of order=k has N=2^k many vertices. Such a tree has depth levels 0 to k. The number of vertices at depth d is the binomial coefficient binom(k,d), hence the name of the tree.
The N=8 example above is order=3 and the number of vertices at each depth is 1,3,3,1 which are binomials (3,0) to (3,3).
A top-down definition is order k tree is two copies of k-1, one at the root and the other a child of that root. In the N=8 order=3 example above, 0-3 is an order=2 and 4-7 is another, with 4 starting as a child of the root 0.
A bottom-up definition is order k tree as order k-1 with a new leaf vertex added to each existing vertex. The vertices of k-1 with an extra low 0-bit become the evens of k. An extra low 1-bit is the new leaves.
Binomial tree order=5 appears on the cover of Knuth "The Art of Computer Programming", volume 1, "Fundamental Algorithms", second edition.
$graph = Graph::Maker->new('binomial_tree', key => value, ...)
The key/value parameters are
N => integer order => integer graph_maker => subr(key=>value) constructor, default Graph->new
Other parameters are passed to the constructor, either
Nis the number of vertices, 0 to N-1. Or instead
ordergives N=2^order many vertices.
Graph::Maker::BalancedTree, if the graph is directed (the default) then edges are added both up and down between each parent and child. Option
undirected => 1creates an undirected graph and for it there is a single edge from parent to child.
The Wiener index of the binomial tree is calculated in
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, "Wiener index of Binomial Trees and Fibonacci Trees", Intl J Math Engg with Comp, 2009. arxiv:0910.4432
For order k it is
(k-1)*4^k + 2^k Wiener(k) = --------------- = 0, 1, 10, 68, 392, ... (A192021) 2
The Wiener index is total distance between pairs of vertices, so the mean distance is, with binomial to choose 2 among the 2^k vertices,
Wiener(k) k MeanDist(k) = ---------------- = k-1 + ------- for k>=1 binomial(2^k, 2) 2^k - 1
The tree for k>=1 has diameter 2*k-1 between ends of the deepest and second-deepest subtrees of the root. The mean distance as a fraction of the diameter is then
MeanDist(k) 1 1 1 ----------- = --- - ------- + ------------------- diameter(k) 2 4*k - 4 (2 - 1/k) * (2^k-1) -> 1/2 as k->infinity
From the bottom-up definition above, a tree of even N has a perfect matching, being each odd n which is a leaf and its even attachment. An odd N has a near-perfect matching (one vertex left over).
The independence number is found by starting with each leaf in an independent set and its attachment vertex not. Per the bottom-up definition this is all vertices. If N odd then the unpaired existing even vertex can be included too, for independence number ceil(N/2).
The domination number similarly, by starting each leaf not in the dominating set and its attachment vertex in the set. Again this is all vertices so domination number floor(N/2), except N=1 domination number 1.
House of Graphs entries for the trees here include
- order=0, https://hog.grinvin.org/ViewGraphInfo.action?id=1310 (single vertex)
- order=1, https://hog.grinvin.org/ViewGraphInfo.action?id=19655 (path-2)
- order=2, https://hog.grinvin.org/ViewGraphInfo.action?id=594 (path-4)
- order=3, https://hog.grinvin.org/ViewGraphInfo.action?id=700
- order=5, https://hog.grinvin.org/ViewGraphInfo.action?id=21088
Entries in Sloane's Online Encyclopedia of Integer Sequences related to these graphs include
A192021 Wiener index
Copyright 2015, 2016, 2017 Kevin Ryde
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