On TD Polynomials of Ring Sum of Graphs

A. R. Latheesh Kumar — Thu, 05 Nov 2026 00:00:00 +0000

Total domination polynomials enumerate the total dominating sets of a graph according to their cardinalities and provide an algebraic means of examining selected graph constructions. This study investigates total domination polynomials for certain graphs obtained through the ring sum, or symmetric difference, operation. The graph families considered are formed from path graphs and cycle graphs together with star graphs. The analysis employs vertex-cover polynomials and the correspondence between total dominating sets of a graph and vertex covering sets of its open-neighbourhood hypergraph. After establishing the relevant definitions and vertex-cover identities, the study derives expressions for the total domination polynomials of ring sums involving odd and even cycles with stars. It then examines ring sums of paths with stars, considering the parity of the path order and the position of the vertex shared with the star. The derivations use decompositions of open-neighbourhood hypergraphs and products of vertex-cover polynomials to obtain the stated expressions. The resulting formulae represent the total domination polynomials of the considered graph families in terms of vertex-cover polynomials of paths and factors arising from the pendant vertices of the star graphs. Within the specified configurations, the study illustrates the use of established vertex-cover methods for analysing total domination in graphs formed through ring sum operations.

Kantorovich Theorem and a Class of Successive Overrelaxation Methods on Error Reduction Factor Per Iteration in the Solution to Nonlinear System of Equations

Stephen Ehidiamhen Uwamusi, Andrew Esaborlupia Uwamusi — Thu, 05 Nov 2026 00:00:00 +0000

This paper provides a basis for the analysis of Kantorovich theorem and a class of Successive Overrelaxation (SOR) methods in the nonlinear system of equations. We provide answers to the question whether a local homeomorphism of a differentiable function is a basis to mean that a Lipeomorphism of signifies a Lipschitz homeomorphism. Using Heinrich and Mankiewicz’s theorem that established every separable metric space is Lipschitz equivalent to a subset of a continuous function, we are led to investigating the matrix splitting as a preconditioner in the iterative process leading to solution of system of nonlinear equations. The emphasis is on the rigours concerning error reduction factor per iteration in the classical iterative methods of Jacobi, Gauss-Siedel, Successive Overrelaxation (SOR) in comparison with Kantorivich theorem. In this regard, we carefully avoided making comments regarding measurements of convergence speeds in terms of iteration matrices for Jacobi, Gauss−Siedel and Successive Overrelaxation methods since SOR has been traditionally known to be the fastest in this class. Further inquiry on performance among this class of linear solvers led to extension of ideas to the cyclic Chevbyshev semi- iterative method in the computation.
The first result we got in the findings is the proof that Newton operator on Jacobi, Gauss−Siedel, Successive overrelaxation (SOR) iterations and Chevbyshev−Semi iteration respectively, satisfied the Keller space.
Secondly, it is established that among these classes of methods , in terms of error reduction per step, it was discovered that Newton−Kantorovich method has the least number of iterations to attain desired accuracy followed closely by Newton −Gauss−Siedel method while Newton−Jacobi iteration maintained a third position with Newton−Chevbyshev −Semi−iterative method coming last in the analysis. Numerical examples given to support our claim proved so. Numerical toolbox used is programmed in Python C++ codes for Matlab routines.

Journal of Advances in Mathematics and Computer Science

On TD Polynomials of Ring Sum of Graphs

Kantorovich Theorem and a Class of Successive Overrelaxation Methods on Error Reduction Factor Per Iteration in the Solution to Nonlinear System of Equations