15 research outputs found
Strong Structural Controllability of Signed Networks
In this paper, we discuss the controllability of a family of linear
time-invariant (LTI) networks defined on a signed graph. In this direction, we
introduce the notion of positive and negative signed zero forcing sets for the
controllability analysis of positive and negative eigenvalues of system
matrices with the same sign pattern. A sufficient combinatorial condition that
ensures the strong structural controllability of signed networks is then
proposed. Moreover, an upper bound on the maximum multiplicity of positive and
negative eigenvalues associated with a signed graph is provided
The zero forcing polynomial of a graph
Zero forcing is an iterative graph coloring process, where given a set of
initially colored vertices, a colored vertex with a single uncolored neighbor
causes that neighbor to become colored. A zero forcing set is a set of
initially colored vertices which causes the entire graph to eventually become
colored. In this paper, we study the counting problem associated with zero
forcing. We introduce the zero forcing polynomial of a graph of order
as the polynomial , where is
the number of zero forcing sets of of size . We characterize the
extremal coefficients of , derive closed form expressions for
the zero forcing polynomials of several families of graphs, and explore various
structural properties of , including multiplicativity,
unimodality, and uniqueness.Comment: 23 page
On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme
We determine the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results.</p