Research Article
The Star Domination Polynomial in Hypergraphs
Issue:
Volume 12, Issue 4, August 2026
Pages:
74-79
Received:
26 June 2026
Accepted:
8 July 2026
Published:
13 August 2026
Abstract: This paper introduces the star domination polynomial of hypergraphs as an extension of the domination polynomial in graphs. Based on the concept of star domination, we define the star domination polynomial as the generating function that counts the star dominating sets of a hypergraph according to their cardinalities. We establish several fundamental properties of this polynomial and investigate its behavior under the disjoint union of hypergraphs. Explicit formulas are obtained for the star domination polynomial of important classes of hypergraphs, including r-uniform complete hypergraphs and sunflower hypergraphs, by characterizing their star dominating sets of different cardinalities. The proposed polynomial provides an algebraic representation of the distribution of star dominating sets and offers a new tool for studying domination in hypergraphs. These results extend the theory of domination polynomials to hypergraphs and provide a foundation for further research on domination parameters, hypergraph invariants, and related combinatorial polynomials.
Abstract: This paper introduces the star domination polynomial of hypergraphs as an extension of the domination polynomial in graphs. Based on the concept of star domination, we define the star domination polynomial as the generating function that counts the star dominating sets of a hypergraph according to their cardinalities. We establish several fundament...
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Research Article
New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability
Issue:
Volume 12, Issue 4, August 2026
Pages:
80-89
Received:
27 February 2026
Accepted:
11 June 2026
Published:
25 August 2026
DOI:
10.11648/j.ijtam.20261204.12
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Abstract: Ostrowski-type integral inequalities provide computable bounds for the error between function values, weighted sums, and integral means, but classical estimates based on a single global derivative bound can be conservative. This study introduces a symmetric 12-step Peano kernel and develops new Ostrowski-type inequalities for functions satisfying three regularity conditions. A fundamental integral identity is established by applying integration by parts over the twelve kernel subintervals. The resulting remainders are bounded using the Gruss, Cauchy-Schwarz, and Diaz-Metcalf inequalities in the relevant function spaces. The sharpest L2 estimate is then applied to cumulative distribution functions on bounded intervals to construct a Certified Expectation Estimator (CEE). The estimator combines a fixed weighted set of CDF evaluations with the L2 norm of the probability density function to approximate the expectation and produce a deterministic, non-asymptotic a priori error bound. Numerical examples involving uniform, Beta(2,2), and truncated normal distributions illustrate that the bound adapts to the density norm and follows the predicted dependence on the interval length. The proposed framework links classical integral inequality theory with certified computation and provides a reproducible method for expectation estimation in applications such as Bayesian inference, reinforcement learning, and neural network verification. The results demonstrate that the symmetric 12-step construction offers a practical balance between analytical tractability, computational cost, and rigorous reliability guarantees.
Abstract: Ostrowski-type integral inequalities provide computable bounds for the error between function values, weighted sums, and integral means, but classical estimates based on a single global derivative bound can be conservative. This study introduces a symmetric 12-step Peano kernel and develops new Ostrowski-type inequalities for functions satisfying t...
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Shama Kochuthundiyil Syed
