Saiful Rahman Mondal

A Cayley graph $\Gamma =\mathrm{Cay}(G,S)$ is a graphical regular representation (GRR) of G if $\mathrm{Aut}(\Gamma )=G$ acts regularly. We call $\Gamma$ a quantum rigid if its quantum automorphism group also satisfies $\mathrm{QAut}(\Gamma )=G$. We prove that a Cayley graph whose coherent (two-dimensional Weisfeiler–Leman) closure is thin is quantum rigid. For Cayley graphs this closure is exactly the S-ring generated by S. The direct in-group refinement algorithm tests thinness in polynomial time; with comparison sorting of the refinement signatures, one round costs $O\left(\right|G{|}^2\log \left|G\right|)$ and at most $\left|G\right|-1$ strict refinement rounds occur, giving the explicit worst-case bound $O\left(\right|G{|}^3\log \left|G\right|)$. Thus thinness supplies a polynomial-time sufficient test for the absence of quantum symmetry, rather than a complexity-theoretic certificate in the NP sense. We also prove that, for a GRR, thinness is equivalent to Schurity of its coherent closure. Hence non-quantum-rigidity of a GRR forces a non-Schurian coherent closure, although the converse is not asserted. Every GRR over a Schur group is therefore quantum rigid. Moreover, a WL-thin Cayley graph is determined up to ordinary isomorphism by its quantum-isomorphism class. Finally, an exact, reproducible census of $\mathrm{193,694}$ connected labelled Cayley graph instances (one for each symmetric connection set, without quotienting by graph isomorphism) in twenty-four finite groups finds no GRR that fails WL-thinness. The accompanying Python 3.14 program contains the group constructions, refinement procedure, exact GRR test and machine-readable output.

In this paper, we introduce and systematically study a new two-parameter family of affine $(q,\eta )$-Gould–Hopper-based Appell polynomials. These polynomials arise by fusing the affine $(q,\eta )$-exponential Gould–Hopper kernel with an invertible Appell multiplier. A central contribution is the affine quasi-monomial construction: an explicit raising operator obtained from a logarithmic-difference quotient of the combined kernel, together with the lowering operator and the resulting q-commutation structure. An explicit double-sum series representation and an affine diffusion equation of order j connecting differences in the two variables, and the full quasi-monomial framework identifying the operator pair $(P_{q,\eta }^{+},P_{q,\eta }^{-})$ are developed. Additional results include a governing difference equation, a converse characterisation, a determinantal representation, an affine addition formula, and an order recursion. Further, the Bernoulli and Euler sub-families are obtained as invertible specialisations, while the Genocchi family is treated separately as a derived noninvertible family through its exact relation with the Euler family. The corresponding structural results are stated with these hypotheses made explicit. Surface plots, numerical value tables, and a numerical illustration of real zeros accompany the theoretical development. The diffusion relation provides a discrete affine analogue of a higher-order evolution equation; no claim of a fully developed physical model is made in the present work.

For $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form $z^n{F}^{″}(z)+a\left(z\right)z^{n-1}{F}^{\prime }\left(z\right)+b\left(z\right)F\left(z\right)+d\left(z\right)=0,n\in \{1,2\},z\in \mathbb{D},$ with normalization $F\left(0\right)=1$, where $a\left(z\right)$, $b\left(z\right)$ and $d\left(z\right)$ are analytic in $\mathbb{D}$. Using advanced techniques from geometric function theory, we generalize and improve existing results, particularly for classes of functions defined by differential equations. Specific applications include generalized Bessel functions, regular Coulomb wave functions, and associated Laguerre polynomials, where we derive improved bounds for their inclusion in lemniscate domains. Additionally, we present open problems, supported by numerical experiments, to guide future research in this direction.

This paper develops a game-theoretic framework for multiple-attribute group decision-making (MAGDM) under an interval-valued intuitionistic fuzzy (IVIF) environment. First, a generalized score function (GSF) is proposed for intuitionistic fuzzy numbers (IFNs) and extended to interval-valued intuitionistic fuzzy numbers (IVIFNs). By incorporating indeterminacy through two flexible parameters, the GSF improves the discrimination and sensitivity of alternative evaluations under uncertain information. Its effectiveness is validated through seven benchmark cases in which the existing score and accuracy functions fail to distinguish the considered IFNs and IVIFNs. In all cases, the GSF yields distinct rankings, resolving the ambiguities. An IVIF MAGDM algorithm is then developed for problems with unknown attribute weights. The aggregated IVIF decision matrix is formulated as the payoff matrix of a two-player zero-sum matrix game (TMG), enabling determination of the attribute weights from assessments of multiple decision-makers. The game models are defined for membership intervals excluding 1 and nonmembership intervals excluding 0, while these boundary cases lie outside the admissible domain of the formulation. An IVIF expected score function is introduced to evaluate the overall performance of alternatives, followed by the GSF to obtain their final ranking. The framework preserves the IVIF structure throughout the decision-making process, with scalarization restricted to the final ranking stage, thereby retaining the inherent uncertainty and hesitation of the original decision assessments. The applicability of the approach is demonstrated through a sustainable air-conditioning system selection problem. Comparative analysis with four existing approaches shows exact agreement with one method and slight differences from the others. Sensitivity analysis with respect to the preference parameter $ \omega $ confirms that the obtained ranking remains unchanged for $ \omega = 0.2 $, $ 0.5 $, and $ 0.8 $, indicating 100% ranking stability. These findings demonstrate the discrimination capability of the GSF and the robustness and reliability of the IVIF MAGDM framework.