On graphs with small ranks: Old and new results

Abstract

The M-rank of a graph G is the rank of the associated matrix M(G). In this paper, we summarize the results about the graphs with a comparatively small M-rank, and propose some problems for further study. In addition, for a real number α ∈ [0, 1], we determine all graphs with Aα-rank 2, where Aα(G) = αD(G) + (1 − α)A(G) and A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of G.