44 research outputs found
The number of -nearly independent vertex subsets
Let be a graph with vertex set and edge set . A subset
of is an independent vertex subset if no two vertices in are
adjacent in . We study the number, , of all subsets of
that contain exactly one pair of adjacent vertices. We call those subsets
1-nearly independent vertex subsets. Recursive formulas of are
provided, as well as some cases of explicit formulas. We prove a tight lower
(resp. upper) bound on for graphs of order . We deduce as a
corollary that the star (the tree with degree sequence
) is the -vertex tree with smallest , while it is
well known that is the -vertex tree with largest number of
independent subsets.Comment: 21 pages, 3 table
Greedy Trees, Subtrees and Antichains
Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root\u27s neighbors, and so on.
They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants.
We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree