12 research outputs found
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 ROOTS OF EDGE COVER POLYNOMIALS OF GRAPHS
AbstractLet G be a simple graph of order n and size m. An edge covering of the graph G is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Let e(G,k) be the number of edge covering sets of G of size k. The edge cover polynomial of G is the polynomial E(G,x)=∑k=1me(G,k)xk. In this paper, we obtain some results on the roots of the edge cover polynomials. We show that for every graph G with no isolated vertex, all the roots of E(G,x) are in the ball {z∈C:|z|<(2+3)21+3≃5.099}. We prove that if every block of the graph G is K2 or a cycle, then all real roots of E(G,x) are in the interval (−4,0]. We also show that for every tree T of order n we have ξR(K1,n−1)≤ξR(T)≤ξR(Pn), where −ξR(T) is the smallest real root of E(T,x), and Pn,K1,n−1 are the path and the star of order n, respectively
Some applications of Wagner's weighted subgraph counting polynomial
We use Wagner's weighted subgraph counting polynomial to prove that the
partition function of the anti-ferromagnetic Ising model on line graphs is real
rooted and to prove that roots of the edge cover polynomial have length at most
. We moreover discuss how our results relate to efficient algorithms for
approximately computing evaluations of these polynomials
ON THE EDGE COVER POLYNOMIAL OF CERTAIN GRAPHS
Let be a simple graph of order and size .The edge covering of is a set of edges such that every vertex of is incident to at least one edge of the set. The edge cover polynomial of is the polynomial,where is the number of edge coverings of of size , and is the edge covering number of . In this paper we study theedge cover polynomials of cubic graphs of order .We show that all cubic graphs of order (especially the Petersen graph) aredetermined uniquely by their edge cover polynomials