We show that the mistake bound for predicting the nodes of an arbitrary weighted graph is characterized (up to logarithmic factors) by the weighted cutsize of a random spanning tree of the graph. The cutsize is induced by the unknown adversarial labeling of the graph nodes. In deriving our characterization, we obtain a simple randomized algorithm achieving the optimal mistake bound on any graph...

Valence-weightings are considered for shortest-path distance moments, as well as related weightings for the so-called “Wiener” polynomial. In the case of trees the valence-weighted quantities are found to be expressible as a combination of unweighted quantities. Further the weighted quantities for a so-called “thorny” graph are considered and shown to be related to the weighted and unweighted q...

In this paper, we present the tool AVERIST, which implements an algorithmic approach for the stability analysis of polyhedral switched systems. It implements an abstraction based model-checking approach proposed in the earlier work of the authors. The tool constructs a weighted graph abstracting the polyhedral hybrid system, and analyzes the graph to infer stability. In the case that the hybrid...

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e., looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin ve...

We classify weighted forests up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces. The word “graph” in this text means a finite undirected graph such that no edge connects a vertex to itself and at most one edge joins any given pair of vertices. A weighted graph is a graph in which each vertex is assigned an integer (called its weight). Two ope...

In the graph balancing problem the goal is to orient a weighted undirected graph to minimize the maximum weighted in-degree. This special case of makespan minimization is NP-hard to approximate to a factor better than 3/2 even when there are only two types of edge weights. In this note we describe a simple 3/2 approximation for the graph balancing problem with two-edge types, settling this very...

The Shot Boundary Detection (SBD) is an early step for most of the video applications involving understanding, indexing, characterization, or categorization of video. The SBD is temporal video segmentation and it has been an active topic of research in the area of content based video analysis. The research efforts have resulted in a variety of algorithms. The major methods that have been used f...

In a network, each arc is assigned a weight. The weight of a path or a cycle is defined as the minimum weight of its arcs. The maximum of weights of all paths between two nodes is defined as the strength of connectedness between the nodes. In network applications, the reduction in the strength of connectedness is more relevant than the total disconnection of the graph. A graph is totally weight...

Let Gw be a weighted graph. The inertia of Gw is the triple In(Gw) = (i+(Gw), i−(Gw), i0(Gw)), where i+(Gw), i−(Gw), i0(Gw) are, respectively, the number of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw) of Gw including their multiplicities. A simple n-vertex connected graph is called a (k − 1)-cyclic graph if its number of edges equals n + k − 2. Let θ(r1, r2, . . . ...

A pebbling move on a weighted graph removes some pebbles at a vertex and adds one pebble at an adjacent vertex. The number of pebbles removed is the weight of the edge connecting the vertices. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The pebbling number of a weighted graph is the smallest number m needed to guarante...