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GRAPHS WHOSE MINIMAL RANK IS TWO WAYNE BARRETT, HEIN VAN DER HOLST , AND RAPHAEL LOEWY
 

Summary: ELA
GRAPHS WHOSE MINIMAL RANK IS TWO
WAYNE BARRETT, HEIN VAN DER HOLST , AND RAPHAEL LOEWY§
Abstract. Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F, G)
be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the
positions corresponding to the edges of G. For example, if G is a path, S(F, G) consists of the
symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in
S(F, G). Then mr(F, G) = 1 if and only if G is the union of a clique with at least 2 vertices and
an independent set. If F is an infinite field such that char F = 2, then mr(F, G) 2 if and only
if the complement of G is the join of a clique and a graph that is the union of at most two cliques
and any number of complete bipartite graphs. A similar result is obtained in the case that F is an
infinite field with char F = 2. Furthermore, in each case, such graphs are characterized as those for
which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is
reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of
a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a
graph that is the union of any number of cliques and any number of complete bipartite graphs. The
number of forbidden subgraphs is now 5, or in the connected case, 3.
Key words. Rank 2, Minimum rank, Symmetric matrix, Forbidden subgraph, Bilinear sym-
metric form.
AMS subject classifications. 05C50, 05C75, 15A03, 15A57.

  

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

 

Collections: Mathematics