We detennine all possible relationa between the height (Weyr) characteristic and the level characteristic of an M-matrix. Under the assumption that the two characteristics have the same number of elements, we detennine the possible relations between the two characteristics for a wider class of matrices, which also contains the class of strictly triangular matrices over an arbitrary field. Given...

We show that the forward and backward propagation can be formulated as a solution of lower and upper triangular systems of equations. For standard feedforward (FNNs) and recurrent neural networks (RNNs) the triangular systems are always block bi-diagonal, while for a general computation graph (directed acyclic graph) they can have a more complex triangular sparsity pattern. We discuss direct an...

Let T,U and V be rings with identity and M be a unitary (T,U)-bimodule, N be a unitary (U, V )bimodule, D be a unitary (T, V )-bimodule . We characterize homomorphisms and isomorphisms of the generalized matrix ring Γ = ( T M D 0 U N 0 0 V )

Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that, for each graph with rare triangular subgraphs, the corresponding quadratic algebra is a Koszul domain with global dimension equal to the number of vertices of the graph.

A walk c1, c2, . . . , cM in an undirected graph G = (V, E) is called a dynamic identifying code, if all the sets I(v) = {u ∈ C : d(u, v) ≤ 1} for v ∈ V are nonempty and no two of them are the same set. Here d(u, v) denotes the number of edges on any shortest path from u to v, and C = {c1, c2, . . . , cM}. We consider dynamic identifying codes in square grids, triangular grids, hexagonal meshes...

Let G be a graph with p vertices and q edges and A = {0, 1, 2, . . . , [q/2]}. A vertex labeling f : V (G) → A induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the indu...

A Hamiltonian path (cycle) of a graph is a simple path (cycle) which visits each vertex of the graph exactly once. The Hamiltonian path (cycle) problem is to determine whether a graph contains a Hamiltonian path (cycle). A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. Supergrid graphs were first introduced by us and include grid grap...

The distance d(f, f ′) between two triangular embeddings f and f ′ of a complete graph is the minimal number t such that we can replace t faces in f by t new faces to obtain a triangular embedding isomorphic to f ′. We consider the problem of determining the maximum value of d(f, f ′) as f and f ′ range over all triangular embeddings of a complete graph. The following theorem is proved: for eve...

A basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. Often the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of size p(v) to each vertex of the triangular lattice so that the s...

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G)  V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investiga...