Locating the Eigenvalues of Trees

On Multiple Eigenvalues of Trees

Let T be a tree of order n > 6 with μ as a positive eigenvalue of multiplicity k. Star complements are used to show that (i) if k > n/3 then μ = 1, (ii) if μ = 1 then, without restriction on k, T has k+ 1 pendant edges that form an induced matching. The results are used to identify the trees with a non-zero eigenvalue of maximum possible multiplicity.

Locating Multiple Gene Duplications through Reconciled Trees

We introduce the first exact and efficient algorithm for Guigó et al.’s problem that, given a collection of rooted, binary gene trees and a rooted, binary species tree, determines a minimum number of locations for gene duplication events from the gene trees on the species tree. We examined the performance of our algorithm using a set of 85 gene trees that contain genes from a total of 136 plant...

Trees with Certain Locating-Chromatic Number

The locating-chromatic number of a graph can be defined as the cardinality of a minimum resolving partition of the vertex set such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex in is expressed in terms of the distances of to all partition classe...

Locating-domination and identifying codes in trees

Locating and total dominating sets in trees

Locating and Total Dominating Sets in Trees by Jamie Marie Howard A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to some vertex in S. In this thesis, we consider total dominating sets of minimum cardinality which have the additional property that distinct vertices of V are totally dominated by distinct subsets of the total dominating set.