Abstract:
An edge labeled graph is a graph G whose edges are labeled with non-zero ideals of a
commutative ring R. A Generalized Spline on an edge labeled graph G is a vertex labeling
of G by elements of the ring R, such that the difference between any two adjacent vertex
labels belongs to the ideal corresponding to the edge joining both the vertices. The set of
generalized splines forms a sub ring of the product ring R|V |
, with respect to the operations
of coordinate-wise addition and multiplication and also becomes a module over the ring
R.This ring which is also a module is known as the generalized spline ring RG, defined on
the edge labeled graph G, for the commutative ring R. We have considered particular
graphs such as complete graphs, complete bipartite graphs and hypercubes, labeling the
edges with the non-zero ideals of an integral domain R and have identified the generalized
spline ring RG for these graphs. Also, general algorithms have been developed to find
these splines for the above mentioned graphs, for any number of vertices and Python code
has been written for finding these splines.We also determine conditions for a subset of
R(G,α) to form a basis for the spline module R(G,α)
, for some classes of graphs such as
Dutch Windmill graph and it’s special cases such as friendship graph,butterfly graph over
GCD domain.We find a generating set of flow-up classes for wheel graphs over the ring
Z/pkZ, where p is prime. Also we classify splines on cycles and wheel graphs over the ring
Z/mZ when m has few prime factors and find a generating set of flow-up classes on these
graphs over Z/mZ. We also determine conditions for a subset of R(G,α) to form a basis of
R(G,α)
for some classes of graphs.We have studied basis criteria for generalized splines on
some isomorphic graphs over GCD domain and constructed flow-up basis for generalized
spline modules on an arbitrary tree.
Description:
A Thesis submitted to Navrachana University Vadodara for the Degree of Doctor of Philosophy in Mathematics: Guide: Dr. Mazumdar Lipika, Researcher: Duggaraju Radhamadhavi, School of Scirnce, Navrachana University, Vadodara, July, 2022