, Volume 65, Issue 3, pp 223-232,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 02 Aug 2011

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

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

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”.