In a graph G with a distribution of pebbles on its vertices, a pebbling move is the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. A weight function on G is a non-negative integer-valued function on the vertices of G. A distribution of pebbles on G covers a weight function if there exists a sequence of pebbling moves that gives a new distribution in which every vertex has at least as many pebbles as its weight. In this paper we give some necessary and some sufficient conditions for a distribution of pebbles to cover a given weight function on a connected graph G. As a corollary, we give a simple formulation for the `weighted cover pebbling number' of a weight function W and a connected graph G, defined by Crull et al. to be the smallest number m such that any distribution on G of m pebbles is a cover for W. Also, we prove a cover pebbling variant of Graham's Conjecture for pebbling.
A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely secure vertex cover and cover pebbling number, is introduced in this paper. The secure vertex cover pebbling number of a graph G is the minimum number m so that every distribution of m pebbles can reach some secure vertex cover of G by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for compl…
A pebbling move on a graph consists of taking two pebbles off from one vertex and add one pebble on an adjacent vertex, the $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on any vertex on $G$ regardless the original distribution of pebbles. Let $ω$ be a positive function on $V(G)$, the $ω$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $ω(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $ω$-cover pebbling number of trees for nonnegative function $ω$, w…
In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics] the authors conjecture that if every vertex of an infinite square grid is reachable from a pebble distribution, then the covering ratio of this distribution is at most $3.25$. First we present such a distribution with covering ratio $3.5$, disproving the conjecture. The authors in the above paper also claim to prove that the covering ratio of any pebble distribution is at most $6.75$. The proof contains some errors. We present a few interesting pebble distributions that this proof does not seem to cover and highlight s…
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
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 $4$. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$cycle covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$cycle. The minimum weight of $k-$cycle covering is the weighted covering number on $k-$cycle, denoted by $τ_{k}(G_{w})$. In this paper, we design a $k-1/2$ approximation algorithm for the weighted covering number on $k-$cycle when $k$ is odd. Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$clique covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$clique. The minimum weight of $k-$cl…
For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.
We consider the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the result of Li and Hou's on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$. Furthermore, we give a formula for the weighted Kirchhoff index of a regular covering of $G$ in terms of that of $G$.
In the classical partial vertex cover problem, we are given a graph $G$ and two positive integers $R$ and $L$. The goal is to check whether there is a subset $V'$ of $V$ of size at most $R$, such that $V'$ covers at least $L$ edges of $G$. The problem is NP-hard as it includes the Vertex Cover problem. Previous research has addressed the extension of this problem where one has weight-functions defined on sets of vertices and edges of $G$. In this paper, we consider the following version of the problem where on the input we are given an edge-weighted bipartite graph $G$, and three positive inte…
A branched covering map of surfaces induces a map in the opposite direction between their arc complexes. We represent a branched covering map combinatorially using what we call a lifting picture, and use this representation to computably solve the membership problem of the set of weighted arc diagrams on a given surface which can be obtained by lifting a weighted arc diagram on a bigon. We provide a brute force solution in the general case, and an efficient solution when the input arc diagram is a triangulation.
Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly chosen from a fixed collection $\mathcal{F}(\mathbb{F}_{q})$ of polynomials in $\mathbb{F}_{q}[x]$ with fixed $m \geq 2$. Under some conditions, these equations are affine models of cyclic $m$-covers of the projective line. Previously, different authors obtained asymptotic results about distributions of points on curves associated to certain collections of polyno…
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $τ$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $τ$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rati…
Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $τ$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; t…
The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted $\ell_1$ norm over the solution set of a linear complementarity problem (LCP). The minimum $\ell_1$ norm of solutions of this LCP is a lower bound on the independent domination number of the graph. Unlike the case of the maximum $\ell_1$ norm, this lower bound is in ge…
Multiparametric families of hypergeometric $τ$-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at ${\bf t} ={\bf 0}$ shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is …
