It is known that the zero forcing number of a graph an upper bound for maximum nullity (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, Fallat, C. Godsil, Haemers, L. Hogben, R. Mikkelson, Narayan, O. Pryporova, I. Sciriha, So, Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wan...

(c) Columns of A are independent. (d) A is tall (i.e., n ≤ m) and full-rank (i.e., rank(A) = min(m,n) = n). Solution: We will show the chain of equivalences (a) =⇒ (b) =⇒ (c) =⇒ (d) =⇒ (a). (a) =⇒ (b): By the rank–nullity theorem, we have dim(N (A)) + rank(A) = n, which implies rank(A) = n (since dim(N (A)) = 0). Since rank(A) = rank(A ), we then have rank(A ) = n. Since rank is equivalent to t...

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...