Abstract
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 \({\frac{r-3}{2}K_2+K_3}\).
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We thank the referees for many important remarks, which lead to a considerable improvement of the manuscript.
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This is one of several papers published together in Designs, Codes and Cryptography on the special topic: “Combinatorics – A Special Issue Dedicated to the 65th Birthday of Richard Wilson”.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Haemers, W.H., Peeters, M.J.P. The maximum order of adjacency matrices of graphs with a given rank. Des. Codes Cryptogr. 65, 223–232 (2012). https://doi.org/10.1007/s10623-011-9548-3
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DOI: https://doi.org/10.1007/s10623-011-9548-3