The weak order polytope P n WO is the polytope in R n(n?1) whose ver-tices correspond to the members of the family of reeexive weak orders on f1; 2 : : : ; ng when each coordinate of R n(n?1) is associated with a diierent ordered pair (i; j) of distinct points in f1; 2; : : : ; ng. The vertex w that corresponds to weak order W has w (i;j) = 1 if i-j in W and has w (i;j) = 0 otherwise, with P n ...

In this paper, we intend to introduce a new heuristic algorithm to apply maximum minimization to Boolean functions with normal SOP form. To implement the proposed algorithm, we use the graph data structure and define the adjacencies. Also, we demonstrate some conditions to achieve the maximum minimization. Through this paper, the problem of shared vertices in more than one adjacency is talked, ...

In this paper, we define and obtain several properties of the (adjacency) energy a hypergraph. particular, bounds for are obtained as functions structural spectral parameters, such Zagreb index radius. We also study how hypergraph varies when vertex/edge is removed or an edge divided. addition, solve extremal problem class hyperstars, show that never odd number.

I study the U.S. Congressional voting record using network theory and computations. I encode the roll call votes from both the House of Representatives and the Senate for 1789-2006 into adjacency matrices (graphs) that encode the extent of agreement between any two legislators in a given House or Senate. I apply concepts such as modularity and various notions of centrality to the adjacency matr...

We propose an approach for data clustering based on optimum-path forest. The samples are taken as nodes of a graph, whose arcs are defined by an adjacency relation. The nodes are weighted by their probability density values (pdf) and a connectivity function is maximized, such that each maximum of the pdf becomes root of an optimumpath tree (cluster), composed by samples “more strongly connected...

We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the grap...

The maximum order of a square (0, 1)-matrix A with a fixed rank r is considered, provided A has no repeated rows or columns. When A is the adjacency matrix of a graph, Kotlov and Lovász [A. Kotlov and L. Lovász. The rank and size of graphs. J. Graph Theory, 23:185–189, 1996.] proved that the maximum order equals Θ(2r/2). In this note, it is showed that this result remains correct if A is symmet...

This code implements imperfect maximum-weight b-matching where b is allowed to vary by node. We describe the reduction to perfect maximum-weight 1-matching, and the API and implementation of this code. Let G = (V,E) be a graph and let W ∈ N|V |×|V | and P ∈ B|V |×|V | denote symmetric weight and adjacency matrices. Let δu denote the degree of node u. Let b ∈ N|V | be a vector such that entry bi...

We propose an image decomposition technique that captures the structure of a scene. An image is decomposed into a matrix that represents the adjacency between the elements of the image and their distance. Images decomposed this way are then classified using a maximum margin regression (MMR) approach where the normal vector of the separating hyperplane maps the input feature vectors into the out...

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element u is before v in the ordering. In the second case, the profit depends on whether u is before v and r is before s. The linear ordering problem is well studied, with ex...