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Linear Algebra and its Applications 429 (2008) 625632 Available online at www.sciencedirect.com
 

Summary: Linear Algebra and its Applications 429 (2008) 625632
Available online at www.sciencedirect.com
www.elsevier.com/locate/laa
Three-connected 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 vertex-set V = {1, 2, . . . , n}, let S(G) be the set of all n n real-valued
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 2-connected graphs with M(G) 2.
In this paper, we show that k-connected graphs G have M(G) k, that k-connected partial k-graphs G
have M(G) = k, and that for 3-connected graphs G, M(G) 3 if and only if G is a partial 3-path.
2008 Elsevier Inc. All rights reserved.
AMS classification: 05C50; 05C83; 15A03; 15A18

  

Source: Argerami, Martin - Department of Mathematics and Statistics, University of Regina

 

Collections: Mathematics