 
Summary: Linear Algebra and its Applications 429 (2008) 625632
Available online at www.sciencedirect.com
www.elsevier.com/locate/laa
Threeconnected graphs whose maximum nullity
is at most three
Hein van der Holst
Department of Mathematics and Computer Science, Eindhoven University of Technology,
5600 MB Eindhoven, The Netherlands
Received 30 May 2007; accepted 24 March 2008
Submitted by S. Fallat
Abstract
For a graph G = (V, E) with vertexset V = {1, 2, . . . , n}, let S(G) be the set of all n × n realvalued
symmetric matrices A which represent G. The maximum nullity of a graph G, denoted by M(G), is the
largest possible nullity of any matrix A S(G). Fiedler showed that a graph G has M(G) 1 if and only
if G is a path. Johnson et al. gave a characterization of all graphs G with M(G) 2. Independently, Hogben
and van der Holst gave a characterization of all 2connected graphs with M(G) 2.
In this paper, we show that kconnected graphs G have M(G) k, that kconnected partial kgraphs G
have M(G) = k, and that for 3connected graphs G, M(G) 3 if and only if G is a partial 3path.
© 2008 Elsevier Inc. All rights reserved.
AMS classification: 05C50; 05C83; 15A03; 15A18
