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Ranks and signatures of adjacency matrices

By S. Akbari, P. J. Cameron and G. B. Khosrovshahi


Two simple operations on graphs (deleting isolated vertices, and identifying vertices with the same neighbour sets) do not change the rank and signature of the adjacency matrix. Moreover, for any given rank, there are only finitely many reduced graphs (those in which distinct vertices have distinct neighbour sets) of any given rank. It follows that any graph parameter which is unchanged by the two reductions (such as clique number or chromatic number) is bounded by a function of the rank. We give a list of some such parameters and best possible bounds in some cases. The rank of a graph is bounded by a function of the number t of negative eigenvalues. Hence the above parameters are also bounded by functions of t. The problem of finding the best possible bound is open. We also report on the determination of all reduced graphs with rank at most 7, and give information of the classification by rank and signature up 1 to rank 7. This also gives (at least implicitly) an exact enumeration of all graphs with rank at most 7. We have also determined the largest reduced graphs of rank 8, and we make a conjecture about the general case. Finally, we discuss some special constructions (line graphs and graph products) from this point of view

Year: 2004
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