Inducibility of topological trees

Abstract

The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Székely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be 'close' to 1/4. We can make this precise by showing that 0.24707 . . . ≤ J(A5) ≤ 0.24745 . . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.

Bounds On The Inducibility Of Double Loop Graphs

“…The inducibility of some family of graphs have been determined, i.e., complete graph K k [19], complete bipartite graph K ρ,ρ ,K ρ,ρ+1 [19] and complete multipartite graphs K k,...,k [17]. Inducibility of various graph classes have also been studied, such as d-ary trees [6], rooted trees [7], directed paths [5], oriented stars [16], net graphs [2] and random Cayley graphs [12]. The precise inducibility of paths of order k ≥ 4 or cycles of order k ≥ 6 are still unknown [11].…”

“…Equation (7) decreases the gap between the upper bound and lower bound of I(C k ) with a multiplicative factor of 2e. This upper bound was later improved by Hefetz and Tyomkyn (Theorem 4.0.2) in [14] for all cycles of size k ≥ 6.…”

Inducibility and universality for trees

“…This concept was introduced for graphs by Pippenger and Golumbic [30]; also see [1,11,17,19,20,24,36,38]. The definition for trees used here is by Bubeck and Linial [3], and it differs slightly from the definition used in [6,7,9,10].…”