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 $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page

Upper bounds for the sum of Laplacian eigenvalues of graphs

AbstractLet G be a graph with n vertices and e(G) edges, and let μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0 be the Laplacian eigenvalues of G. Let Sk(G)=∑i=1kμi(G), where 1⩽k⩽n. Brouwer conjectured that Sk(G)⩽e(G)+k+12 for 1⩽k⩽n. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for Sk(G), and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs