| 摘要: | For a graph G on n vertices and a field F. the minimum rank of G over F, written as mr(F)(G), is the smallest possible rank over all n x n symmetric matrices over F whose (i, j)th entry (for i not equal j) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M(F)(G) = n - mr(F)(G). The minimum rank problem of a graph G is to determine mr(F)(G) (or equivalently, M(F) (G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioaba, D. Cvetkovic S.M. Fallat, C. Godsil, W. Haemers, L Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovic, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M(F) (G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G) = M(F) (G) for some field F? This paper focuses on exploring the above question. (C) 2010 Elsevier Inc. All rights reserved. |
