 
Summary: Linear Algebra and its Applications 409 (2005) 1331
www.elsevier.com/locate/laa
On the difference between the maximum
multiplicity and path cover number for treelike
graphs
Francesco Barioli a, Shaun Fallat b,,1, Leslie Hogben c
aSchool of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6
bDepartment of Mathematics and Statistics, University of Regina, Regina, Sask., Canada S4S 0A2
cDepartment of Mathematics, Iowa State University, Ames, IA 50011, USA
Received 17 May 2004; accepted 21 September 2004
Available online 11 November 2004
Submitted by S. Kirkland
We dedicate this work to Pauline van den Driessche for her life long contributions to linear algebra and
her support of the linear algebra community
Abstract
For a given undirected graph G, the maximum multiplicity of G is defined to be the largest
multiplicity of an eigenvalue over all real symmetric matrices A whose (i, j)th entry is non
zero whenever i /= j and {i, j} is an edge in G. The path cover number of G is the minimum
number of vertexdisjoint paths occurring as induced subgraphs of G that cover all the vertices
of G. We derive a formula for the path cover number of a vertexsum of graphs, and use
