 
Summary: Linear Algebra and its Applications 392 (2004) 289303
www.elsevier.com/locate/laa
Computation of minimal rank and path cover
number for certain graphs
Francesco Barioli a, Shaun Fallat a,1, Leslie Hogben b,
aDepartment of Mathematics and Statistics, University of Regina, Regina,
Saskatchewan, Canada S4S 0A2
bDepartment of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011, USA
Received 22 December 2003; accepted 22 June 2004
Submitted by R.A. Brualdi
Abstract
For a given undirected graph G, the minimum rank of G is defined to be the smallest
possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever
i /=j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertex
disjoint paths occurring as induced subgraphs of G that cover all the vertices of G. For trees,
the relationship between minimum rank and path cover number is completely understood.
However, for nontrees only sporadic results are known. We derive formulae for the minimum
rank and path cover number for graphs obtained from edgesums, and formulae for minimum
rank of vertex sums of graphs. In addition we examine previously identified special types of
vertices and attempt to unify the theory behind them.
