Summary: Linear Algebra and its Applications 375 (2003) 111
Graphs whose positive semi-definite matrices
have nullity at most two
Hein van der Holst
Matematicko-Fyzikalni Fakulta, Katedra Aplikovane Matematiky, Univerzita Karlova v Praze,
Malostranske Nam 25, 118 00 Praha 1, Czech Republic
Received 3 July 2000; accepted 17 June 2003
Submitted by V. Mehrmann
Let G = (V, E) be a undirected graph containing n vertices, and let MG be the set of all
Hermitian n × n matrices M = (mi,j ) with mi,j /= 0 if i and j are connected by one edge of
G, with mi,j C if i and j are connected by at least two edges, with mi,j = 0 if i /= j, and i
and j are not connected by an edge of G, and with mi,i for i = 1, . . . , n a real number. What
is the largest nullity attained by any positive semi-definite matrix M MG?
In this paper we characterize, for t = 1 and 2, those graphs G for which the maximum
nullity is not greater than t.
© 2003 Elsevier Inc. All rights reserved.
Keywords: Positive semi-definite; Nullity; Symmetric matrices; Graph structure