84 research outputs found
Cordial Digraphs
A -labeling of a set is said to be friendly if the number of elements
of the set labeled 0 and the number labeled 1 differ by at most 1. Let be a
labeling of the edge set of a graph that is induced by a labeling of the
vertex set. If both and are friendly then is said to be a cordial
labeling of the graph. We extend this concept to directed graphs and
investigate the cordiality of directed graphs. We show that all directed paths
and all directed cycles are cordial. We also discuss the cordiality of oriented
trees and other digraphs
(0,1)-Matrices, Discrepancy and Preservers
Let m and n be positive integers, and let R = (r1, . . . , rm) and S = (s1, . . . , sn) be nonnegative integral vectors. Let A(R,S) be the set of all m × n (0, 1)-matrices with row sum vector R and column vector S. Let R and S be nonincreasing, and let F(R) be the m × n (0, 1)-matrix, where for each i, the ith row of F(R,S) consists of ri 1’s followed by (n−ri) 0’s. Let A ∈ A(R,S). The discrepancy of A, disc(A), is the number of positions in which F(R) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when m = n, the transpose mapping
A note on k-primitive directed graphs
AbstractWe consider the problem of which primitive directed graphs can be k-colored to yield a k-primitive directed graph. If such a k-coloring exists, then certainly such a graph must have at least k cycles. We prove that any primitive directed graph admits a 2-coloring that is 2-primitive. By contrast, for each k⩾4, we construct examples of primitive directed graphs having k cycles for which no k-coloring is k-primitive. We also give some partial results for the case that k=3
Linear operators strongly preserving r-potent matrices over semirings
AbstractA matrix X is said to be r-potent if Xr=X. We investigate the structure of linear operators on matrices over antinegative semirings that map the set of r-potent matrices into itself and the set of matrices which are not r-potent into itself
The characterization of operators preserving primitivity for matrix k-tuples
AbstractWe obtain a complete characterization of surjective additive operators acting on the Cartesian product of several matrix spaces over an antinegative semiring without zero divisors, which map primitive matrix k-tuples to primitive matrix k-tuples
A characterization of linear operators that preserve isolation numbers
We obtain characterizations of Boolean linear operators that preserve some of the isolation numbers of Boolean matrices. In particular, we show that the following are equivalent: (1) preserves the isolation number of all matrices; (2) preserves the set of matrices with isolation number one and the set of those with isolation number for some ; (3) for , preserves matrices
with isolation number , and those with isolation number , (4) maps to and preserves the set of matrices of isolation number 2; (5) is a -operator, that is, for fixed permutation matrices and , matrix ~ or, and where is the transpose of
Linear preservers of term ranks of matrices over semirings
AbstractThe term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we study linear operators that preserve term ranks of matrices over S. In particular, we show that a linear operator T on matrix space over S preserves term rank if and only if T preserves term ranks 1 and α(≥2) if and only if T preserves two consecutive term ranks in a restricted condition. Other characterizations of term-rank preservers are also given
Linear Operators That Preserve Two Genera of a Graph
If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations
(2,3)-Cordial Oriented Hypercubes
L. B. Beasley recently defined a digraph labeling called (2,3)-cordial.
Digraphs for which a (2,3)-cordial labeling can be applied are called
(2,3)-cordial digraphs. Herein, we consider the existence and identification of
(2,3)-cordial oriented hypercubes. We demonstrate that for every nonzero
dimension, there exists a (2,3)-cordial oriented hypercube. Additionally, we
demonstrate that not all oriented hypercubes of nonzero dimension are
(2,3)-cordial. Finally, we present preliminary results regarding the
identification of (2,3)-cordial oriented hypercubes, particularly for dimension
3.Comment: 14 pages, 6 figure
- …