Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 223–232

The maximum order of adjacency matrices of graphs with a given rank

Open Access


We look for the maximum order m(r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m(r) = 2(r+2)/2 − 2 if r is even, and m(r) = 5 · 2(r−3)/2 − 2 if r is odd. We prove the conjecture and characterize G in the case that G contains an induced subgraph \({\frac{r}{2}K_2}\) or \({\frac{r-3}{2}K_2+K_3}\).


Graph Adjacency matrix 

Mathematics Subject Classification (2000)

05B20 05C50 


  1. 1.
    Akbari S., Cameron P.J., Khosrovshahi G.B.: Ranks and signatures of adjacency matrices (preprint 2004).
  2. 2.
    Godsil C.D., Royle G.F.: Chromatic number and the 2-rank of a graph. J. Combin. Theory Ser. B 81, 142–149 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Kotlov A., Lovász L.: The rank and size of graphs. J. Graph Theory 23, 185–189 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Econometrics and Operations ResearchTilburg UniversityTilburgThe Netherlands

Personalised recommendations