Gallai-Edmonds Structure Theorem for Weighted Matching Polynomial

In this paper, we prove the Gallai-Edmonds structure theorem for the most general matching polynomial. Our result implies the Parter-Wiener theorem and its recent generalization about the existence of principal submatrices of a Hermitian matrix whose graph is a tree. keywords:Comment: 34 pages, 5 figure

Evolutionary trees: an integer multicommodity max-flow-min-cut theorem

In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem

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

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