1 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