In the paper triangular graphs are discussed. The class of triangular graphs is of special interest as unifying basic features of complete graphs with trees and being used on many various occasions. They model processor networks in which local computations can be easily implemented. The main issue addressed in the paper is to characterize class of triangular graphs (defined globally) by local m...

We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota– Baxter anti-homomorphism from general regularized functionals on the Feynman graph Hopf algebra to triangular matrices with entries in a Rota–Baxter algebra. For characters mapping to the group of unipotent triangular matr...

In 1995, Brouwer proved that the toughness of a connected k-regular graph G is at least k/λ − 2, where λ is the maximum absolute value of the non-trivial eigenvalues of G. Brouwer conjectured that one can improve this lower bound to k/λ − 1 and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to k/λ. In this pape...

A systematic procedure for developing the unified signal flow graph model of a fuzzy PI controller is presented in this paper. From this unified model it is possible to determine the complete behavior of the Fuzzy PI controller. As a first step in constructing the signal flow graph the analytical structures oflinear fuzzy PI controller are needed. In view of this by considering triangular/trape...

Let R be a noncommutative ring. The zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R)− {0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, there is an edge x → y if and only if xy = 0. In this paper we investigate the zero-divisor graph of triangular matrix rings over commutative rings. Mathematics Subject Classification: 16S70; ...

Linear codes that can be obtained from designs associated with the complete graph onnvertices and its line graph, the triangular graph, are examined. The codes have length n choose 2, dimension n or n − 1, and minimum weight n− 1 or 2n− 4. The parameters of the codes and their automorphism groups for any odd prime are obtained and PD-sets inside the symmetric group Sn are found for full permuta...

An open-locating-dominating set (OLD-set) is a subset of vertices of a graph such that every vertex in the graph has at least one neighbor in the set and no two vertices in the graph have the same set of neighbors in the set. This is an analogue to the well-studied identifying code in the literature. In this paper, we prove that the optimal density of the OLD-set for the infinite triangular gri...

The anti-Kekulé number is the smallest number of edges that must be removed from a connected graph with a perfect matching so that the graph remains connected, but has no perfect matching. In this paper the values of the Anti-Kekulé numbers of the infinite triangular, rectangular and hexagonal grids are found, and they are, respectively, 9, 6 and 4.

To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A−1 have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer invers...