#### Date of Defense

Summer 4-13-1995

#### Department

Mathematics

#### First Advisor

Gary Chartrand, Mathematics and Statistics

#### Second Advisor

Michelle Schultz, Mathematics and Statistics

#### Third Advisor

Dawn Jones, Mathematics and Statistics

#### Abstract

Two edges in a graph are independent if they are not adjacent. A set of edges is independent if every two edges in the set are independent. A set of independent edges in a graph *G* is also called a matching in *G*. The greatest number of independent edges in a graph *G* is denoted by β_{1}(*G*). A matching of cardinality β_{1}(*G*) is called a maximum matching in *G*. The number βˉ_{1}(*G*) is the minimum cardinality among the maximal matchings of *G*. Bounds are established for β_{1}(*G*) in terms of βˉ_{1}(*G*); namely, for every nonempty graph *G*, βˉ_{1}(*G*)≤β_{1}(*G*)≤2βˉ_{1}(*G*). Furthermore, for every two positive integers *a* and *b* with *a*≤*b*≤2*a*, there exists a graph *G* of order 2*b* such that βˉ_{1}(*G*)=*a* and β_{1}(*G*)=*b*. Moreover, the graph *G* with this property can be required to be connected or even 2-connected. It is also shown that for every graph *G* and every integer *k* with βˉ_{1}(*G*)˂*k*˂β_{1}(*G*), there exists a maximal matching in *G* with *k* edges. Sharp upper and lower bounds are presented for βˉ_{1}(*G*)+βˉ_{1}(*G*ˉ). The maximum matchings of a graph *G* can themselves be represented by a graph. The matching graph *M*(*G*) of a graph is that graph whose vertices are the maximum matchings of *G* and two vertices *M*_{0} and *M*_{1} are adjacent in *M*(*G*) if *M*_{0} and *M*_{1} differ by one edge. A graph H is a matching graph if there is a graph *G* such that *H*=*M*(*G*). The problem of determining which graphs are matching graphs is studied. Several graphs are shown to be matching graphs, and it is also shown that not all graphs are matching graphs.

#### Recommended Citation

Roehm, Denny James, "Edge Idependence in Graphs" (1995). *Honors Theses*. 296.

https://scholarworks.wmich.edu/honors_theses/296

#### Access Setting

Honors Thesis-Campus Only