he basis number, b(G), of a graph G is defined to be the least integer k such that G has a k-fold basis for its cycle space. In this paper we investigate the basis number of the composition of theta graphs with stars
The basis number of a graph G is defined to be the least positive integer d such that G has a d-fold basis for the cycle space of G. We investigate the basis number of the cartesian product of stars and wheels with ladders, circular ladders and Möbius ladders.
We generalise the familiar notions of invariant basis number, rank condition, stable finiteness and strong rank condition
from rings to modules. We study the inter relationship between these properties, identify various classes of modules possessing
these properties and investigate the effect of many standard module theoretic operations on each one of these properties.
We also tackle the important problem of preservation or non-preservation of these properties when we pass respectively to
the module of polynomials, power series or inverse polynomials.
We first tackle certain basic questions concerning the Invariant Basis Number (IBN) property and the universal stably finite
factor ring of a direct product of a family of rings. We then consider formal triangular matrix rings and obtain information
concerning IBN, rank condition, stable finiteness and strong rank condition of such rings. Finally it is shown that being
stably finite is a Morita invariant property.