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Nowhere-zero and Zero-sum Flows in Graphs and Designs

Ghodrati, Amirhossein | 2019

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 51752 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saieed
  7. Abstract:
  8. A flow for a graph or a combinatorial design is an element of the nullspace of its incidence matrix. For an integer r ≥ 2, an r-flow is a flow whose values belong to {±1;±2; : : : ;±(r - 1)} and for a real number r ≥ 2, a circular r-flow is a flow whose values are real numbers in the set [-(r- 1),-1]U[1, r - 1].In the context of undirected graphs, flows are called zero-sum flows. In the first part of this dissertation, we study circular zero-sum flows of graphs and prove a necessary and sufficient condition for the existence of a circular zero-sum r-flow in a graph in terms of its factors. Also, we investigate the minimum value of r for which a k-regular graph has a circular r-flow and obtain some lower and upper bounds for this number.In the next part, we study bases for the nullspace of the incidence matrices of a directed or undirected graph consisting of r-flows and prove the existence of such a basis for some values of r and some families of graphs.The last part of the dissertation is devoted to the study of bases consisting of r-flows for the nullspace of the incidence matrices of complete designs.Also, we prove a result on the dimension of the linear space spanned by the characteristic vectors of all Steiner systems with given parameters
  9. Keywords:
  10. Zero-sum Flow ; Nowhere Zero Flow ; Circular Flow ; Steiner Triple System (STS) ; Complete Design

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