The number of 11-nearly independent vertex subsets

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of $\sigma_1$ are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on $\sigma_1$ for graphs of order $n$. We deduce as a corollary that the star $K_{1,n-1}$ (the tree with degree sequence $(n-1,1,\dots,1)$) is the $n$-vertex tree with smallest $\sigma_1$, while it is well known that $K_{1,n-1}$ is the $n$-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