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

More trees with large energy and small size

In a previous paper [E. O. D. Andriantiana, MATCH Commun. Math Comput. Chem. 68 (2012) 000–000] trees with a fixed number n of vertices were ordered according to their energy, and a large number of trees with greatest energy were characterized. These results, however, hold only if n is large enough. We now analyze the energy–ordering of trees for small values of n (up to 100) and establish the first few greatest–energy species. The results obtained for small values of n significantly differ from those valid for large values of n.Publishe

The number of independent subsets and the energy of trees

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.ENGLISH ABSTRACT: See full text for abstractAFRIKAANSE OPSOMMING: Sien volteks vir opsommin

Maximum Wiener Index of Trees With Given Segment Sequence

A segment of a tree is a path whose ends are branching vertices (vertices of degree greater than 2) or leaves, while all other vertices have degree 2. The lengths of all the segments of a tree form its segment sequence. In this note we consider the problem of maximizing the Wiener index among trees with given segment sequence or number of segments, answering two questions proposed in a recent paper on the subject. We show that the maximum is always obtained for a so-called quasi-caterpillar, and we also further characterize its structure

Energy and related graph invariants

Thesis (PhD)--Stellenbosch University, 2013.Please refer to full text to view abstract