The Enumeration of Minimum Path Covers of Trees
Sher, Merielyn
Sher, Merielyn
Abstract
A path cover of a tree T is a collection of induced paths of T that are vertex disjoint and cover all the vertices of T. A minimum path cover (MPC) of T is a path cover with the minimum possible number of paths, and that minimum number is called the path cover number of T. A tree can have just one or several MPC's. Prior results have established equality between the path cover number of a tree T and the largest possible multiplicity of an eigenvalue that can occur in a symmetric matrix whose graph is that tree. We hope to gain insights into the different ways that maximum multiplicity occurs among the multiplicity lists of T by enumerating its MPC's. The overall strategy is to divide and conquer. Given any tree T, several techniques are introduced to decompose T into smaller components. Then, the number of MPC's of these smaller trees can be calculated and recombined to obtain the number of MPC's for the original tree T.
Description
Date
2022-04-01