 
Summary: COLLOQUIUM
Shahla Nasserasr
University of Regina
The Minimum Rank of
Universal Adjacency
Matrices
Friday, January 27, 2012
3:30 p.m.
RIC 209
Abstract: For a simple graph G on n vertices, a matrix of the form U(a, b, c, d) =
aA + bI + cJ + dD, where A is the 0,1adjacency matrix of G, J is the all ones
matrix of size n, I is the identity matrix of size n, and D is the diagonal matrix
with the degrees of the vertices in the main diagonal, and a (not zero), b, c, d
are scalars, is called a universal adjacency matrix of G. An analogous parameter
to the minimum rank of a given graph G is the minimum rank over all matrices
in the set of universal adjacency matrices of G. This parameter is called the
minimum universal rank of G, and is denoted by mur(G). Graphs with mur(G)
equal to zero and one are characterized. The minimum universal rank of some
families of graphs such as complete graphs, complete bipartite graphs, paths and
cycles are presented. A formula for the minimum universal rank of a regular graph
