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On 1-sum flows in undirected graphs

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.13001/1081-3810.3003
  3. Abstract:
  4. Let G = (V,E) be a simple undirected graph. For a given set L ⊂ ℝ, a function ω: E → L is called an L-flow. Given a vector γ ∈ ℝv, ω is a γ-L-flow if for each υ ∈ V, the sum of the values on the edges incident to υ is γ(υ). If γ(υ) = c, for all υ ∈ V, then the γ-L-flow is called a c-sum L-flow. In this paper, the existence of γ-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L*:= L {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum ℝ*-flow or a 1-sum ℤ*-flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow
  5. Keywords:
  6. Bipartite graph ; c-sum flow ; L-flow ; γ-L-flow
  7. Source: Electronic Journal of Linear Algebra ; Volume 31, Issue 1 , 2016 , Pages 646-665 ; 10813810 (ISSN)
  8. URL: http://repository.uwyo.edu/ela/vol31/iss1/46