Motivated by the notion of chip-firing on dual graph a planar graph, we consider ‘integral flow chip-firing’ an arbitrary G. The rule is governed $${\mathcal {L}}^*(G)$$ , Laplacian G determined choosing basis for lattice integral flows We show that any admits such so M-matrix, leading to firing these elements avalanche finite. This follows from more general result bases lattices may be indepen...

In this paper, we extend the de nition of topological labeling of nite directed acyclic graphs to in nite acyclic periodic graphs. Unlike nite acyclic graphs, we show that acyclic periodic graphs cannot be topologically labelled with integers in general. However, we show that these graphs can be labelled with rationals. We also give su cient conditions for the existence of an integral topologic...

We prove a Riemann-Roch theorem for real divisors on edgeweighted graphs over the reals, extending the result of Baker and Norine for integral divisors on graphs with multiple edges.

We prove a Riemann-Roch theorem for real divisors on edge-weighted graphs over the reals, extending the result of Baker and Norine for integral divisors on graphs with multiple edges.

We study a well-known linear programming relaxation of the p-median problem. We give a characterization of the directed graphs for which this system of inequalities defines an integral polytope. As a consequence, we obtain that the p-median problem is polynomial in that class of graphs. We also give an algorithm to recognize these graphs. © 2011 Elsevier B.V. All rights reserved.

A sum graph is a graph for which there is a labeling of its vertices with positive integers so that two vertices are adjacent if and only if the sum of their labels is the label of another vertex. Integral sum graphs are defined similarly, except that the labels may be any integers. These concepts were first introduced by Harary, who provided examples of such graphs of all orders. The family of...

We characterize the graphs for which a linear relaxation of a facility location problem defines a polytope with all integral extreme points. We use a transformation to a stable set problem in perfect graphs. Based on this transformation, these graphs can be recognized in polynomial time. © 2014 Elsevier B.V. All rights reserved.