On order and rank of graphs

Abstract

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most m(r)=2(r+2)/2−2 if r is even and m(r)=5·2(r−3)/2−2 if r is odd. In this article, we prove that if the conjecture is not true, then there would be a counterexample of rank at most 46. We also show that every reduced graph of rank r has at most 8m(r)+14 vertices.

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Correspondence to Ali Mohammadian.

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Ghorbani, E., Mohammadian, A. & Tayfeh-Rezaie, B. On order and rank of graphs. Combinatorica 35, 655–668 (2015). https://doi.org/10.1007/s00493-015-2922-4

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Mathematics Subject Classification (2010)

  • 05C50
  • 15A03