Focused helium ion beam (He-FIB) irradiation with 30 keV ions is a key tool for nanoscale patterning and defect engineering in high transition temperature (Tc) cuprate superconducting devices, yet its usable fluence window is constrained by the competing requirements of reliable superconductivity suppression and minimal structural degradation. In this work, we provide a comprehensive dataset on the effects of large-area 30 keV He+ ion exposure on the electric transport and superconducting properties of epitaxial YBa2Cu3O7 (YBCO) thin films. X-ray diffraction shows a fluence-driven loss of crystalline order accompanied by an out-of-plane lattice expansion and an orthorhombic-to-tetragonal transition, culminating at predominant amorphization at the highest fluence of 1 x 10^16 cm^-2. Raman spectra exhibit increasing disorder while lacking signatures of oxygen depletion, indicating that irradiation mainly generates oxygen-related Frenkel defects rather than changing the carrier concentration. Consistently, with increasing fluence, the normal-state resistivity ρ_N(T) at temperature T above Tc increases strongly, while d ρ_N/d T remains nearly unchanged at moderate fluence. The suppression of Tc is accurately described by Abrikosov-Gor'kov pair breaking and reaches complete quenching of superconductivity at 4.5 x 10^15 cm^-2. The anisotropic upper critical fields decrease approximately exponentially with increasing fluence, the vortex activation energy is reduced, and the anisotropy drops, in contrast to oxygen-depleted YBCO. Hall-angle analysis confirms a nearly constant carrier density but a systematic increase in defect scattering and reduced mobility, consistent with a crossover toward the dirty limit at high fluence. These results establish quantitative fluence thresholds and a practical operational window for He-FIB nanopatterning of YBCO quantum circuits.

Background: Understanding whether patients follow similar or distinct patterns of care is important for characterizing clinical practice, identifying patient subgroups, and supporting quality improvement. However, routine healthcare trajectories are difficult to compare directly because patients may differ in their diagnostic workup, treatment sequencing, timing of clinical events, and documentation practices. Despite this variation, trajectories often contain recurring patterns at the cohort level. Methods: To address this challenge, we present a framework that explicitly separates cohort-level typical pathway identification from patient-level inference. At the cohort level, we derive an interpretable representation of care processes using a rule-based algorithm to identify typical healthcare pathways, resulting in a compact pathway graph. These pathways are then modeled as Markov chains and used as structured components in an admixture model, allowing each patient to be represented as a probabilistic mixture of typical pathways rather than being assigned to a single pathway component. The resulting admixture weights provide a compact representation of patient trajectories for subgroup characterization. We further assess the stability of the identified pathways and inferred admixture representations across multiple train-test splits. Results: Across train-test splits, the framework demonstrated consistent pathway structures and patient-level mixture patterns. Applied to routine care data from prostate cancer patients undergoing radical prostatectomy, the framework identified interpretable care patterns and supported the identification of patient subgroups with similar clinical event patterns. Conclusion: Overall, the proposed framework provides an interpretable and stable approach for summarizing treatment pathways and characterizing patient subgroups in real-world practice.

The precise structural characterization of lattice coverings remains a profound open problem in the geometry of numbers, notably lacking an analogue to Minkowski's classical criterion for lattice packings. To bridge this gap, we introduce and structurally classify a novel geometric framework: \textit{minimal covering bodies}. First, we establish a constructive lattice covering criterion in three dimensions based on the classical Kuhn triangulation. Furthermore, while classical space-tiling parallelohedra are limited to finitely many combinatorial types, we prove that minimal covering bodies can exhibit infinitely many distinct types in both asymmetric and symmetric higher-dimensional settings. Finally, we propose a Minkowski-type geometric criterion and an algebraic intersection framework, aiming to reduce the continuous three-dimensional covering problem to a computationally verifiable discrete condition.

Dual-species atom interferometry in space promises precise tests of the Universality of Free Fall (UFF), with a sensitivity that grows quadratically with the extended interrogation time accessible in weightlessness. These tests demand exquisite control over the expansion energies of both condensed sources as well as over their differential center-of-mass dynamics. We propose a trap-quenched collimation technique featuring in-trap excitations of collective modes compatible with state-of-the-art atom-chip setups. Using NASA's Cold Atom Laboratory aboard the International Space Station, we demonstrate it on a single-species $^{87}$Rb condensate. By controlling the center-of-mass release dynamics we observe free expansion times up to 700 ms and measure a two-dimensional expansion energy of $k_B \cdot 78\pm 9 \;\mathrm{pK}$ in the imaging plane. A detailed model of the magnetically-induced dynamics indicates that this corresponds to a two-dimensional expansion energy of about $k_B \cdot 15^{+12}_{-5}\; \mathrm{pK}$ along two of the condensate's eigenaxes. Finally, we theoretically study this trap-quenched collimation scheme for a $^{41}$K-$^{87}$Rb mixture, predicting a simultaneous collimation that meets the expansion energy requirements for a state-of-the-art UFF test at the $10^{-15}$ accuracy level.

In this paper, we introduce and study the notions of (n,Q)-ideals ans phi-(n,Q)-ideals of commutative rings.

Fluorescent labeling is a cornerstone of DNA visualization and a key enabler of random access in DNA-based data storage. However, the stochastic nature of biochemical processes, including synthesis, hybridization, and optical readout, induces \emph{burst} synchronization errors within the resulting labeling sequences. To address this critical challenge, we formally introduce \emph{burst $t$-deletion/insertion $\mathcal{A}$-labeling codes,} designed to correct a single burst of $t$ deletions or insertions in the label domain. Our contributions are threefold. \begin{itemize} \item \textbf{Fundamental limit.} We establish an information-theoretic lower bound of $\log_4 n + \mathcal{O}(1)$ on the redundancy of any such code for all $t \ge 1$ with $t \mid n$. To the best of our knowledge, this resolves the first information-theoretic lower bound even for the single-error case \(t=1\). \item \textbf{Explicit construction.} For $t \ge 2$, $t \mid n$, and $n \ge 7t + 3$, we propose explicit encoding and decoding algorithms, both running in $\mathcal{O}(n^2)$ time. A novel generalized Run-Length Limited (RLL) constraint is introduced to bridge the structural mismatch between the DNA encoding domain and the label error domain. \item \textbf{Asymptotic optimality.} The proposed scheme achieves redundancy $\log_4 n + (t-1)\log_4 \log_{8/3} n + \mathcal{O}(1)$, matching the dominant term of the lower bound up to a small $\mathcal{O}(\log\log n)$ overhead, rendering the construction asymptotically optimal for fixed $t$. \end{itemize}

The solar corona has an extremely low density, and its brightness is only about one millionth of that of the photosphere. High-dynamic-range imaging of its faint structure is therefore essential for studying coronal heating, coronal mass ejections, and space weather. Quantitative coronagraph imaging requires flat-field measurement and calibration, which underpin intensity calibration, small-scale feature detection, and long-term cyclic analysis. This paper analyzes the coronagraph imaging chain and the origins of flat-field errors, including optical aberrations, stray light, and pixel-response non-uniformity, and summarizes the resulting calibration requirements of next-generation coronagraphs. On this basis, ground-based and space-based flat-fielding methods are systematically reviewed: the ground-based methods include integrating-sphere uniform light sources, opal glass/diffuser plates, clear-sky and thin-cloud backgrounds, and solar-disk scanning, while the space-based methods include internal light sources and diffuser plates, attitude-roll and off-corona offset observations, and multi-phase statistical self-consistent flat-fielding. Their accuracy, resource cost, and applicability are compared. The review shows that no single method is simultaneously high-precision, easy to update, and engineer-friendly; a hierarchical, multi-method calibration framework is therefore recommended. Finally, a new method is proposed in which lithographically generated structured light fields, combined with Fourier-optics and machine-learning inversion, are used to estimate the pixel-response function. Preliminary experiments show that this method achieves a lower residual error than the integrating-sphere and opal-glass methods, providing a high-precision reference for future wide-band, high-resolution coronagraph calibration.

Interactions between boiling bubbles are well known to influence departure dynamics and heat transfer, yet their role in governing nucleation stability, whether sites activate reproducibly, persist, and deactivate under changing thermal loads, remains poorly understood. Here we show that nucleation can be a collective process: neighboring sites at close spacings exhibit reduced variability and sustained activity, consistent with a non-local hydrodynamic shielding mechanism whereby neighboring bubbles slow the intervening flow, reducing convective heat removal and stabilizing vapor embryos. To isolate near-wall nucleation dynamics from bubble-scale vapor removal, we design surfaces comprising pairs of cavities, with intra-pair spacing tuned to the boundary layer scale and inter-pair separation to the departure diameter scale. While the former governs nucleation behavior, the latter governs collective vapor removal once sites are fully active, yielding transitions between excessive, promotive, and isolated departure regimes. Together these results establish a multiscale framework for designing robust, high-performance boiling surfaces.

Cycle-level DRAM simulators provide accurate and flexible models for DRAM and memory controller operations and enable research on current and future memory systems. Therefore, they are critical for improving the performance, efficiency, and robustness of DRAM-based memory systems. Ramulator 2.0 (successor of Ramulator) is a highly modular and extensible cycle-accurate DRAM simulator that enables rapid exploration of new ideas in DRAM-based memory systems. A MICRO 2024 best paper runner-up publication, A Mess of Memory System Benchmarking, Simulation and Application Profiling, which we refer to as "the Mess paper," with all three artifact badges awarded (including "Reproducible"), proposes a new benchmark to evaluate real and simulated memory system performance. While doing so, it makes strong negative claims about Ramulator 2.0 and shows unexpected results. In this talk and the associated extended abstract, we demonstrate that these results and claims in the Mess paper are incorrect and are due to configuration and simulator usage errors made in the Mess paper. We describe four best practices to aid users and developers of simulation tools to avoid such issues in the future. We emphasize the importance of contacting simulator authors and developers when unexpected results are observed (especially and importantly before publishing such results), to ensure these simulators are used with correct configurations and as intended. Our investigation also aims to stimulate discussion on artifact evaluation practices and on mechanisms for correcting results and artifacts after publication. To aid future works and reproduction of all our results, we open source all our code and scripts at https://github.com/CMU-SAFARI/Cleaning-up-the-Mess. We refer the reader to our full ISPASS 2026 paper and its artifact for the complete analysis, detailed methodology, and extended results.

Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $Δ\ge 2$ and $p \ge n^{-1/Δ- \varepsilon_H} (\log n)^{ω(1)}$ for an explicit $\varepsilon_H >0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the naïve mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log\log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

Internal water waves arise when there is a change in density stratification in a fluid, which may occur in an oceanographical context due to variations in temperature, salinity, or other fluctuations in the equations of state. We present a derivation of nonlinear integrable models for the propagation of interfacial internal waves arising between two fluid layers of different densities (at the so called pycnocline). We examine the integrable Benjamin-Ono (BO) equation as an internal wave model, incorporating underlying currents by permitting a sheared current in both fluid layers. The BO equation arises for a specific small-amplitude asymptotic regime. We show that the BO soliton characteristics are strongly affected by the shear current parameters.

This work investigates how toxin-mediated interactions and directed movements shape the emergence of coherent structures in plant-herbivore systems. The analysis focuses on a two-compartment model enclosing a toxin-dependent functional response and a cross-diffusion term that represents ecologically plausible herbivores' movement towards, or away from, vegetation. Two distinct dynamical regimes arise depending on toxicity strength. Under weak toxicity, the system admits at most one biologically feasible coexistence equilibrium, which may lose stability through a Hopf bifurcation generating small-amplitude temporal oscillations. Under strong toxicity, the nonlinear functional response becomes non-monotonic, allowing for multiple coexistence equilibria and abrupt regime shifts. The influence of cross-diffusion on stability is also examined, identifying the conditions under which Turing instabilities and mixed spatiotemporal patterns occur. Near the corresponding bifurcation thresholds, Stuart-Landau amplitude equations are derived via weakly nonlinear analysis, providing a unified framework for the modulation of oscillatory, stationary, and combined Turing-Hopf modes. Numerical simulations corroborate the theoretical predictions, illustrating transitions from spatially uniform states to oscillations, spatial patterns, and mixed behaviour. Overall, this manuscript highlights how chemical defences, nonlinear feedbacks, and movement strategies jointly determine the emergence, selection, and robustness of coherent structures in plant-herbivore systems.

We carry out continuum quantum Monte Carlo calculations of the quantum-mechanical Wigner distribution functions of selected nuclei, up to $^{16}$O. These distributions provide a form of quantum tomography of the spatial and momentum structure of the system. They also help identify the location of high-momentum regions in atomic nuclei and provide insight into the onset of alpha clustering. Besides their intrinsic interest, these distributions will be useful for neutrino event generators, as they correlate the positions and momenta of nucleons in the initial target state. To facilitate their application, we address the need to store them compactly by developing an accurate Gaussian Process emulator that automatically preserves their normalization.

We investigate Private Information Retrieval (PIR) in the context of synthetic DNA-based data storage. While PIR is a well-studied primitive for digital databases, extending it to DNA-based databases presents unique challenges arising from biochemical query mechanisms and their complexity. We propose two approaches for adapting two-server PIR protocols to DNA-based storage, balancing privacy, efficiency, and feasibility. These approaches illustrate how information-theoretic privacy trade-offs manifest in DNA-based storage systems.

Blockchain interoperability enables independent blockchain systems to communicate and exchange assets across heterogeneous networks. However, the lack of comprehensive security mechanisms remains a critical weakness -- one that attackers have already exploited to cause hundreds of millions of dollars in asset losses. This paper presents a systematic identification and classification of security threats facing interoperable blockchain systems, along with corresponding countermeasures for each. We organize threats into five categories: (1) core blockchain attacks, (2) network attacks, (3) interoperability-specific attacks, (4) social engineering, and (5) code vulnerabilities, with particular attention to smart contract weaknesses. For each identified threat, we analyze its attack surface and propose effective defensive strategies. The resulting taxonomy provides a structured foundation for designing and evaluating secure blockchain interoperability solutions.

Efficient generation of entangled photons typically relies on spontaneous parametric down-conversion (SPDC) in phase-matched macroscopic nonlinear media. However, generating entanglement under phase-matching constraints requires additional bulk optics or interferometers. In contrast, ultrathin van der Waals semiconductors - such as transition metal dichalcogenides (TMDs) - exhibit strong enough optical nonlinearities for SPDC to be observed from subwavelength-thick media, thereby bypassing conventional phase-matching constraints. In this microscopic domain, the intrinsic crystal symmetry governs the nonlinear optical response, enabling the native generation of polarization-entangled photon pairs. However, generating these states efficiently has been fundamentally restricted by the material's coherence length ($L_c$), which limits the attainable conversion efficiency. Here, we investigate periodically-poled TMDs (PPTMDs) designed to scale up this interaction via quasi-phase matching. We demonstrate that mechanically flipping the sign of the nonlinearity at precise intervals of $L_c$ introduces quasi-phase matching, that scales the pair-production rate while preserving the pristine, symmetry-generated polarization entanglement, with fidelities exceeding 99%. Backed by a rigorous theoretical model, our work clarifies the interplay between crystal symmetry and propagation effects in thin nonlinear media, providing a new avenue for engineering quantum light in nanophotonic systems.

In this paper, we introduce maximal green sequences of infinite length in abelian length categories and establish natural bijections among infinite maximal green sequences, finest stability data, and complete backward $\operatorname{Hom}$-orthogonal (briefly, CBHO) sequences of bricks. For cluster quiver patterns (equivalently, cluster algebras) of infinite rank, we define maximal green sequences of infinite length via $c$-vectors. Moreover, we prove that under suitable conditions such a maximal green sequence in a cluster quiver pattern of infinite rank can be categorified as a maximal green sequence of torsion classes in the corresponding representation category via a specific construction, thereby providing an example of a maximal green sequence in an abelian length category. This framework establishes a connection between infinite maximal green sequences in representation theory and maximal green sequences for cluster algebras of infinite rank.

The Laplacian spectral radius of a graph is the largest eigenvalue of its Laplacian matrix. Previously, upper bounds for the Laplacian spectral radius were proposed using a backward-reconstruction procedure starting from expressions equal to $2x$ and substituting local degree data. A numbered list of 68 such candidate bounds was subsequently investigated, resulting in the refutation of 30 of these bounds; two additional bounds were later refuted in a separate study. This paper updates the status of the remaining 36 candidate bounds. Of these remaining bounds, we confirm 22 and refute 12, leaving only two upper bounds open. The valid bounds follow primarily from classical Laplacian spectral radius bounds and the Collatz--Wielandt comparison; the refutations are carried out through explicit counterexamples relying on equitable partitions.

Higher-dimensional theories of gravity predict that gravitational waves (GWs) can propagate into extra spatial dimensions, leading to modified amplitude damping over cosmological distances. Measurements of GW luminosity distances therefore provide a unique probe of the dimensionality of spacetime. In this work, we constrain higher-dimensional GW propagation using the dark-siren method with the Gravitational-Wave Transient Catalog 4.0 (GWTC-4). We adopt a phenomenological parameterization motivated by braneworld scenarios, in which deviations from General Relativity are characterized by the spacetime dimension number $D$ and a crossover scale $R_c$ governing the transition between four- and higher-dimensional gravity. We perform a hierarchical Bayesian analysis combining 141 compact binary coalescences from GWTC-4 with line-of-sight galaxy information from the GLADE+ catalog. For a prior $H_0 \in [65,77]\ {\rm km~s^{-1}Mpc^{-1}}$ and $\log(R_c/{\rm Mpc}) \in [2.7,4.0]$, we obtain $D = 4.38^{+1.91}_{-1.01}$ (68\% credible interval). We also find that the inferred posterior distribution of $R_c$ accumulates near the upper prior boundary, indicating that the crossover scale remains poorly constrained by current observations. We further show that the inferred constraint on $D$ depends sensitively on the assumed prior range of $R_c$, which determines the characteristic distance scale at which deviations from General Relativity become significant. Our results provide the first GWTC-4 dark-siren constraints on higher-dimensional GW propagation and demonstrate that current observations remain consistent with four-dimensional General Relativity.

Finite element simulations of structures with nonlinear material behavior require advanced material models to provide accurate predictions. However, the computational costs of these models can be high, as they solve coupled differential algebraic equations at each integration point, in each equilibrium iteration, in every time step. In this study, we propose a machine learning-based framework to accelerate these computations by explicitly calculating the state variable updates with neural networks, enabling large time steps with low computational costs. The neural networks operate on invariants, and the necessary and sufficient evolution directions are determined analytically based on the training data. Furthermore, the proposed framework enforces exact fulfillment of the plastic consistency condition. To evaluate the proposed framework, a prototype model with the von Mises yield criterion and nonlinear kinematic hardening is chosen. Only 10 cycles of multiaxial proportional loading are used to generate the training data. After evaluating the proposed framework in material point simulations, we incorporate it into finite element simulations to evaluate its accuracy and computational efficiency in a boundary value problem. The results from both material point and finite element simulations show a very promising numerical performance of the neural network-based time integrator. It provides very good accuracy and numerical stability, as well as a noticeable gain in computational time for a single strain increment per load segment.

A large body of experimental studies of swift heavy ion latent tracks in dielectric materials has produced a wide range of estimates of track size. We investigate the physical meaning of these estimates by examining the different criteria of track boundary probed by various experimental techniques, including SAXS, XRD, chemical etching and conductometry. We show that different methods probe different physical aspects of ion induced modification, such as electron density redistribution, molecular ordering, chemical reactivity and charge separation, resulting in different determinations of effective track boundaries. Particular attention is paid to polymer films with electret-like properties, where post irradiation redistribution of weakly bound electrons may play an important role in the evolution of latent track structure.

We apply Lüscher's method to the vector channel of the scattering amplitude of Pseudo-Nambu-Goldstone-Bosons (PNGBs), in the $Sp(4)$ lattice gauge theory coupled to $N_f=2$ flavors of Wilson-Dirac fundamental fermions. We generalize existing algorithms and numerical implementations of the method, to adapt them to this prominent candidate for the completion of proposed extensions of the Standard Model (SM). We present the first ab initio measurements of key properties of the vector resonances in the theory, including the coupling to the PNGBs, that are relevant to direct and indirect new physics searches, both for composite Higgs models (CHMs), as well as for strongly interacting massive particle (SIMP) realizations of dark matter. We also present a global update of the spectroscopy of the mesons in the theory, improving both the statistics and analysis systematics in respect to previous lattice measurements reported in the literature.

Optically active spin defects are foundational for quantum technologies, yet common approximations underestimate their internal conversion (IC) rates by orders of magnitude. We propose a broad, predictive framework to compute IC rates that incorporates multi-configurational effects via many-body wavefunctions in TDDFT, and includes all-phonon-mode contributions via analytical non-adiabatic couplings. Our approach resolves discrepancies with experiment, achieving quantitative agreement for the NV$^-$ center in diamond, and identifying a previously overlooked non-radiative channel in the divacancy triplet lifetime in SiC.

The predictive approach to Bayesian inference accesses the posterior distribution via a sequence of one-step-ahead predictives, enabling inference via predictive resampling without Markov chain Monte Carlo. In the random-design regression setting, an explicit specification of the predictive design distribution is required, yet the impact of this choice has received little formal attention. We study the role of this predictive design distribution in parametric martingale posteriors for regression, and identify predictive notions of identifiability and design invariance that are essential for valid inference, particularly in the high-dimensional regression setting. Building on these foundations, we introduce a novel class of parametric martingale posteriors for regression that satisfies a weak form of these desiderata, and naturally accommodates the high-dimensional setting through regularization. We then illustrate our method through a simulation.

Minimising bacterial surface adhesion and subsequent biofilm formation in industrial and medical settings requires understanding how bacteria are transported and adhere to complex surface geometries in the presence of non-uniform flow. In this paper, we consider the transport of a dilute suspension of motile bacteria through a corrugated two-dimensional channel with perfectly adhesive walls. We asymptotically analyse the diffusive boundary layer that forms in high velocity flows using a curvilinear coordinate system based on the fluid streamfunction, presenting a similarity solution to the diffusivity-varying diffusion-type equation that arises. From this solution, we derive an analytical expression for the bacterial adhesion rate as a function of surface arclength and the spatially varying wall shear rate. Our model predicts that bacterial adhesion becomes localised on curved surfaces, with bacteria showing preferential adhesion to wall `peaks' at lower shear rates and preferential adhesion to wall `valleys' at higher shear rates. More broadly, our results highlight how spatially varying flows generated by complex geometries can lead to localised bacterial adhesion, with potential implications for both enhancing and minimising biofilm formation.

A central problem in statistical physics is to extend the Boltzmann distribution to nonequilibrium steady states (NESS). We prove that NESS on large dense networks have Boltzmann-like occupation despite extensive entropy production. We further show that the active-matter heuristic of "low rattling" is asymptotically exact. Intuitively, these NESS spend a greater fraction of their time in states they leave more slowly. This explanation extends to the broader class of "equiaccessible" steady states, which play a role in our analysis akin to that of equilibrium in linear response.

We provide a complete proof that Yang--Baxter bi-vector deformations are solution generating transformations in Type II supergravity. The proof does not rely on the assumption that the vielbein is invariant under Lie derivative along all Killing vectors entering the bi-vector. We analyse transformation of boundary terms and derive a simple expression for the case, when the boundary does not change its geometrical locus.

In this paper, we study the community detection problem in the stochastic block model (SBM) under privacy constraints. We introduce private and highly efficient algorithms for exact community detection within the SBM framework. Our algorithms represent the first differentially private methods capable of achieving exact recovery in a wide range of model parameters with near-linear time and space complexity. This is a significant improvement over previous SBM recovery algorithms, which either required pseudo-polynomial time or a quadratic scaling of resources for a constant privacy budget. Central to our approach is the introduction of a new concept, adaptive disjoint-star algorithms. These algorithms efficiently explore the graph's structure by querying node degrees on edge-disjoint subgraphs. We demonstrate that this general class of algorithms inherently offers strong privacy guarantees, a result that potentially holds value beyond the scope of SBM community detection. Finally, in we perform an empirical analysis of our algorithms showing that they can scale exact recovery on graphs with two orders of magnitude more nodes than prior work.

MaxSTN and MinSTN -- proposed by Papamanthou and Tollis (TCS 2008, JGAA 2010) -- are two algorithms for producing $st$-orientations of biconnected graphs with long and short longest paths respectively. Based on extensive experiments on planar and non-planar graphs of up to 5,000 nodes, it was conjectured that $\ell_{\max} \geq \ell_{\min}$ for every biconnected graph $G$, where $\ell_{\max}$ and $\ell_{\min}$ denote the longest-path lengths of the two orientations. This paper disproves this conjecture by exhibiting a biconnected graph on 9 vertices for which MaxSTN yields $\ell_{\max}=6$ while MinSTN yields $\ell_{\min}=7$, regardless of how ties are broken in either algorithm.

This paper resolves several open questions regarding the algebraic and metric inheritance of distance-based regularities under graph products, with a particular focus on Commutative Distance Degree-Regular (CDDR) graphs. We characterize the behavior of CDDR, distance mean-regular (DMR), and distance degree-regular (DDR) graphs under the strong and direct (tensor) products. To address the non-closure of classical regularities under the direct product, we introduce and study a novel, broader family of graphs: \textit{Distance Walk Regular (DWR)} graphs. Furthermore, we relax the global commutativity condition by defining and structuralizing the class of $i$-CDDR graphs, providing deep characterizations within the realm of Cayley graphs. By developing systematic lifting operations, we construct infinite families of graphs with arbitrarily large order and diameter within specific structural intersections, effectively settling topological gaps in the current regularity hierarchy.