The number of maximum matchings in a tree

Abstract

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) m(T) of a tree T of given order. While the trees that attain the lower bound are easily characterised, the trees with the largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order n is at most O(1.391664n) (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupień on the number of maximal matchings (maximal with respect to set inclusion).

Null decomposition of trees

“…This result are all direct consequences of the Sachs theorem, see Theorem 3.8, pag.31 in Bapat (2014). Determine the family of n-trees that maximize mpT q, the number of maximum matching in a tree T , is a hard problem solved in see Heuberger and Wagner (2011). The third corollary is a new way to think about this interesting problem: The number of maximum matching only depends on the S-set.…”

Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

“…It is worthwhile to investigate how far the dual behaviour goes as the papers [26] and [28] generated considerable interest in different disciplines [4,6,10,11,12,13,14,17,20,21,23,24,31,34,38]. The analogy certainly goes one step further, to describe the "middle part" of the tree.…”

Stepwise target controllability identifies dysregulations of macrophage networks in multiple sclerosis

“…Control network theory, linking network structure to dynamics through linear or nonlinear models, has been shown to be a more principled approach for identifying driver nodes in an interconnected system (Rugh & Kailath, 1995 ; Sontag, 1998 ). While theoretically these approaches can give a minimum set of driver nodes sufficient to steer the system into desired states, their exhaustive identification might be difficult in practice as there exists in general a very large number of equivalent controllable walks, even in relatively simple networks (Heuberger & Wagner, 2011 ; S. G. Wagner, 2007 ). In the case of criteria based on the manipulation of controllability matrices (Hautus, 1970 ; Kalman, 1963 ), the presence of many walks can for example induce numerical errors due to the different orders of magnitude in the matrix elements.…”

“…While this approach elegantly solves numerical issues, it can nevertheless not tell which configuration, among all the possible ones, is the most relevant. In general, there is a factorial number of equivalent configurations (with the same number of inputs), and enumerating all possible matchings (Uno, 1997 ) rapidly becomes unfeasible, even for simple graphs such as trees (Heuberger & Wagner, 2011 ; S. G. Wagner, 2007 ), bipartite graphs (Y. Liu & Liu, 2004 ), or random graphs (Zdeborová & Mézard, 2006 ). Thus, the research of alternative strategies to characterize the candidate driver nodes is crucial for the concrete application of network controllability tools.…”