In this paper the properties of node-node adjacency matrix in acyclic digraphs are considered. It is shown that topological ordering and node-node adjacency matrix are closely related. In fact, first the one to one correspondence between upper triangularity of node-node adjacency matrix and existence of directed cycles in digraphs is proved and then with this correspondence other properties of ...

we study the set of all determinants of adjacency matrices of graphs with a given number of vertices. using brendan mckay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. using an idea of m. newman, it is proved that if $...

In this paper, we propose a cluster-adjacency based method to order sentences for multi-document summarization tasks. Given a group of sentences to be organized into a summary, each sentence was mapped to a theme in source documents by a semi-supervised classification method, and adjacency of pairs of sentences is learned from source documents based on adjacency of clusters they belong to. Then...

Estimation of multiple directed graphs becomes challenging in the presence of inhomogeneous data, where directed acyclic graphs (DAGs) are used to represent causal relations among random variables. To infer causal relations among variables, we estimate multiple DAGs given a known ordering in Gaussian graphical models. In particular, we propose a constrained maximum likelihood method with noncon...

In 1995 T. Matsui considered a special family 0/1-polytopes for which the problem of recognizing the non-adjacency of two arbitrary vertices is NP-complete. In 2012 the author of this paper established that all the polytopes of this family are present as faces in the polytopes associated with the following NP-complete problems: the traveling salesman problem, the 3-satisfiability problem, the k...

A method of naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph has been previously suggested. A polyhedron can be built using its name. Classes of convex n-acra (i.e. n-vertex polyhedra) are strictly (without overlapping) ordered by their names. In this paper the relationship between the Fedorov algorithm to generate the whole combinatorial vari...

Gene cluster significance tests that are based on the number of genes in a cluster in two genomes, and how compactly they are distributed, but not their order, may be made more powerful by the addition of a test component that focuses solely on the similarity of the ordering of the common genes in the clusters in the two genomes. Here we suggest four such tests, compare them, and investigate on...

Visualization of the adjacency matrix enables us to capture macroscopic features a network when elements are aligned properly. Community structure, consisting several densely connected components, is particularly important feature and structure can be identified through it close block-diagonal form. However, classical ordering algorithms for matrices fail align such that community visible. In t...

The elimination tree is a rooted tree that is computed from the adjacency graph of a symmetric matrix A. The height of the elimination tree is one restricting factor when solving a sparse linear system Ax = b on a parallel computer using Cholesky factorization. An eecient algorithm is presented for the problem of ordering the nodes in a tree G so that its elimination tree is of minimum height. ...