J. Richard Lundgren

Food Webs, Competition Graphs, Competition-Common Enemy Graphs, and Niche Graphs

This paper surveys the recent work on competition graphs of food webs and some new graphs related to competition graphs, namely, competition-common enemy graphs and niche graphs. Also investigated are digraphs having interval competition graphs, and a partial solution to this problem for a class of (i, j)-competition graphs is given. Several open problems related to these graphs as well as generalized competition graphs are mentioned.

Niche graphs

If D = (V, A) is an acyclic digraph and G = (V, E) is a graph such that two vertices x and y are adjacent in G if and only if they have a common predator vertex or prey vertex in D, then G is called a niche graph. It is easy to show that not all graphs are niche graphs. However in many cases it is possible to adjoin a finite set of vertices, say I,, to the vertex set Vof both G and D, and also some additional arcs to the arc set to obtain G’ and D’ respectively where G’ is a niche graph, V’ = VU 1, and E’= E. The smallest number of vertices that one must adjoin to G to obtain a niche graph is called the niche number of G. Some classes of niche graphs are investigated, including paths and cycles. We also calculate the niche number of some other graphs. An infinite class of graphs is exhibited in which none of the graphs in that class has a niche number and a characterization of niche graphs is given.

The Domination and Competition Graphs of a Tournament

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M1 and M2 of M(G) are adjacent if and only if |M1 - M2| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs.

A characterization of graphs of competition number m

Abstract In this note we give a characterization of graphs with competition number less than or equal to m. We also give an alternate proof of a theorem characterizing competition graphs.

Inverting graphs of rectangular matrices

Abstract This paper addresses the question of determining the class of rectangular matrices having a given graph as a row or column graph. We also determine equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix. In connection with these graph inversion problems we discuss the concept of minimal inverses. This concept turns out to have two different forms in the case of one-graph inversion. For the two-graph case we present a method of determining when an inverse is minimal. Finally we apply the two-graph theorem to a class of energy related matrices.

Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices

We study the minimum number of complete bipartite subgraphs needed to cover and partition the edges of a k-regular bigraph on 2n vertices. Bounds are determined on the minima of these numbers for fixed n and k. Exact values of the minima are found for all n and k ≤ 4. The same results hold for directed graphs. Equivalently, we have determined bounds on the minimum value of the Boolean and nonnegative integer ranks of binary n × n matrices with constant row and column sum k for fixed n and k, obtaining the exact values of the minimum for k ≤ 4.

Graph Theoretic Methods for the Qualitative Analysis of Rectangular Matrices

In the past few years, many large models including several energy models have been represented by rectangular matrices, and graphs appear to be valuable in investigating connectivity and other properties of these models. It is the purpose of this paper to establish some of the basic foundations for the use of graphs and digraphs to investigate properties of rectangular matrices. A variety of graphs and digraphs associated with rectangular matrices are introduced, and several theorems related to connectivity and tearing are proved. There are also a few applications to the area of computer-assisted analysis.

Competition Graphs of Strongly Connected and Hamiltonian Digraphs

A radio communication network can be modeled by a digraph,

Interval competition graphs of symmetric digraphs

D

( i,j ) competition graphs

, where there is an arc from vertex