Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 223–232

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

Open AccessArticle

DOI: 10.1007/s10623-011-9548-3

Cite this article as:
Haemers, W.H. & Peeters, M.J.P. Des. Codes Cryptogr. (2012) 65: 223. doi:10.1007/s10623-011-9548-3


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 \({\frac{r-3}{2}K_2+K_3}\).


GraphAdjacency matrix

Mathematics Subject Classification (2000)

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© The Author(s) 2011

Authors and Affiliations

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