Jacobus H. Koolen

Basic Phylogenetic Combinatorics

Phylogenetic combinatorics is a branch of discrete applied mathematics concerned with the combinatorial description and analysis of phylogenetic trees and related mathematical structures such as phylogenetic networks and tight spans. Based on a natural conceptual framework, the book focuses on the interrelationship between the principal options for encoding phylogenetic trees: split systems, quartet systems and metrics. Such encodings provide useful options for analyzing and manipulating phylogenetic trees and networks, and are at the basis of much of phylogenetic data processing. This book highlights how each one provides a unique perspective for viewing and perceiving the combinatorial structure of a phylogenetic tree and is, simultaneously, a rich source for combinatorial analysis and theory building. Graduate students and researchers in mathematics and computer science will enjoy exploring this fascinating new area and learn how mathematics may be used to help solve topical problems arising in evolutionary biology.

Characterizing distance-regularity of graphs by the spectrum

We characterize the distance-regular Ivanov-lvanov-Faradjev graph from the spectrum, and construct cospectral graphs of the Johnson graphs, Doubled Odd graphs, Grassmann graphs, Doubled Grassmann graphs, antipodal covers of complete bipartite graphs, and many of the Taylor graphs. We survey the known results on cospectral graphs of the Hamming graphs, and of all distance-regular graphs on at most 70 vertices.

Equiangular lines in Euclidean spaces

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d = 14 and d = 16 . Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.

Krein parameters and antipodal tight graphs with diameter 3 and 4

We determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q114 vanishes.

Delsarte clique graphs

In this paper, we consider the class of Delsarte clique graphs, i.e. the class of distance-regular graphs with the property that each edge lies in a constant number of Delsarte cliques. There are many examples of Delsarte clique graphs such as the Hamming graphs, the Johnson graphs and the Grassmann graphs. Our main result is that, under mild conditions, for given s>=2 there are finitely many Delsarte clique graphs which contain Delsarte cliques with size s+1. Further we classify the Delsarte clique graphs with small s.

The Tight Span of an Antipodal Metric Space: Part II—Geometrical Properties

Abstract Suppose that X is a finite set and let ℝX denote the set of functions that map X to ℝ. Given a metric d on X, the tight span of (X,d) is the polyhedral complex T(X,d) that consists of the bounded faces of the polyhedron [ P(X,d) := {f ∈ ℝX : f(x)+f(y) ≥ d(x,y)}. ] In a previous paper we commenced a study of properties of T(X,d) when d is antipodal, that is, there exists an involution σ: X→ X: x↦

Six Points Suffice

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An algorithm for computing virtual cut points in finite metric spaces

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There are Finitely Many Triangle-Free Distance-Regular Graphs with Degree 8, 9 or 10

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On electric resistances for distance-regular graphs

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