The inverse eigenvalue problem for probe graphs
arxiv(2024)
摘要
In this paper, we initiate the study of the inverse eigenvalue problem for
probe graphs. A probe graph is a graph whose vertices are partitioned into
probe vertices and non-probe vertices such that the non-probe vertices form an
independent set. In general, a probe graph is used to represent the set of
graphs that can be obtained by adding edges between non-probe vertices. The
inverse eigenvalue problem for a graph considers a family of matrices whose
zero-nonzero pattern is defined by the graph and asks which spectra are
achievable by matrices in this family. We ask the same question for probe
graphs. We start by establishing bounds on the maximum nullity for probe graphs
and defining the probe graph zero forcing number. Next, we focus on graphs of
two parallel paths, the unique family of graphs whose (standard) zero forcing
number is two. We partially characterize the probe graph zero forcing number of
such graphs and prove some necessary structural results about the family.
Finally, we characterize probe graphs whose minimum rank is 0, 1, 2, n-2, and
n-1.
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