Abstract Based on R. Leussu et al.’s data, a tabulation of the epoch of the first sunspot group in each hemisphere (“ t 1−H ”) in Cycles 8–24 was recently performed by J. G. Richard. The hemispheric-average first-group emergence epoch (“ t 1−2H ”) was found to be, as a median, 1.4 yr before sunspot-number minimum (“tSNmin”). Now, in the nine even-numbered cycles, a strong linear anticorrelation (Pearson coefficient r = −0.89) is found between sunspot-number maximum (“SNmax”) and the lead time from t 1−2H to SNmax epoch. Using, instead, the lead time from tSNmin as M. Waldmeier did in 1935 (“Waldmeier effect”), r = −0.84 only. Thus, the effect detected in 1935 appears to physically set in as early as t 1−2H in even cycles: a “first-spot effect.” Neither effect is significant at the 95% confidence level in odd cycles.

An improved result on the stability of odd cycles

Intelligent diagnosis of hot-rolled strip steel convexity faults based on hierarchical attention cycle graph networks and causal inference

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of these pieces. From this, we prove an $e$-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.

Social computing systems rely on environmental and behavioral inputs for providing fair processing in a wide range of application support. The input is fetched as text, audio, observation, etc., in which natural language processing is being applied in recent days. Voice-based actuation and processing in these systems result in uncertain events, increasing computational tardiness. This paper introduces a graph convolutional network-based feature processing (GCN-FP) model for addressing the above-mentioned issue. The network augments the voice/audio input features like a connected cyclic graph for succeeding computations. In this process, the training and recurrent graph mapping and disconnection are performed using the certainty factor. This certainty factor relies on the mapped and exhausted features observed in a single graph iteration. Based on the single mapping, decisions for social computing systems are provided, with controlled errors. The features that are not involved in the cyclic process or remain unmapped are identified as errors, and hence, they are included in the training part. Therefore, the proposed scheme achieves better recommendation accuracy, processing ratio, computation complexity, and uncertainty factor.

Abstract The 11-yr variation of galactic cosmic-ray flux lags behind the variation of the sunspot number. An average ~1-yr time-lag is expected from the outward propagating solar wind with the frozen-in photospheric magnetic field varying in the solar cycle, and from the inward diffusive transport of cosmic-ray particles. The long-term neutron monitor data, however, show that the time-lag is significantly longer (shorter) in the odd (even) solar cycle. In this paper, we analyze the time-lag in proton and electron fluxes observed by the CALET. It is found that the time-lag is similar in proton and electron fluxes during an A > 0 polarity epoch of the solar dipole magnetic field. In an even solar cycle 24 including a polarity reversal from A < 0 to A > 0, on the other hand, it is found that the time-lag of proton (electron) flux variation is significantly shorter (longer) than the average ~1-yr lag by analyzing the combined data with CALET and AMS-02. This is the first observation of the charge-sign dependent time-lag. We demonstrate that these observations can be qualitatively interpreted in terms of different 11-yr time profiles of proton and electron fluxes in A > 0 and A < 0 epochs expected from the drift effect.

Abstract The 11-yr variation of galactic cosmic-ray flux lags behind the variation of the sunspot number. An average ~1-yr time-lag is expected from the outward propagating solar wind with the frozen-in photospheric magnetic field varying in the solar cycle, and from the inward diffusive transport of cosmic-ray particles. The long-term neutron monitor data, however, show that the time-lag is significantly longer (shorter) in the odd (even) solar cycle. In this paper, we analyze the time-lag in proton and electron fluxes observed by the CALET. It is found that the time-lag is similar in proton and electron fluxes during an A > 0 polarity epoch of the solar dipole magnetic field. In an even solar cycle 24 including a polarity reversal from A < 0 to A > 0, on the other hand, it is found that the time-lag of proton (electron) flux variation is significantly shorter (longer) than the average ~1-yr lag by analyzing the combined data with CALET and AMS-02. This is the first observation of the charge-sign dependent time-lag. We demonstrate that these observations can be qualitatively interpreted in terms of different 11-yr time profiles of proton and electron fluxes in A > 0 and A < 0 epochs expected from the drift effect.

The unit graph of a commutative ring with a non-zero identity is a graph with vertices as ring elements, and there is an edge between two distinct vertices if their sum is a unit. This study investigates the decomposition of the unit graph by examining its induced subgraphs and analyze key graph invariants, such as connectivity, diameter, and girth, for a finite local ring. We further decompose the unit graph of certain finite commutative rings into fundamental structures, such as cycle and star graphs.

ABSTRACT Let be a finite, loopless graph that may contain multiedges. We call a ring graph if is obtained from a cycle by duplicating some edges. Denote by and the chromatic index and maximum degree of , respectively. Kőnig's classical result implies that if is a bipartite graph. Goldberg showed that , where is the length of a shortest odd cycle of . Stiebitz, Scheide, Toft, and Favrholdt conjectured that if reaches this upper bound, then contains a ring graph as a subgraph with the same chromatic index. Cao, Chen, He, and Jing found some counterexamples for the conjecture. In this paper, we establish a necessary and sufficient condition for the conjecture of Stiebitz et al. to hold. More specifically, writing with in the division‐remainder form, we show that if then the conjecture holds, otherwise the conjecture fails. If graph has an odd number of vertices, a matching of covering all, but one, vertices of is called a near‐perfect‐matching of . We characterize ring graphs, as well as ‐graphs, that have a near‐perfect‐matching factorization, and use these decomposition theorems to obtain the above characterization of the truth of the conjecture of Stiebitz, Scheide, Toft, and Favrholdt.

This paper extends the spectral analysis of distance-based matrices associated with chemical graph structures of order n, focusing on the distance matrix D(G), the distance Laplacian DL(G), and the distance signless Laplacian DQ(G). We investigate the spectral integrality of these matrices for selected acyclic and cyclic hydrocarbon molecular graphs by examining whether their corresponding spectra consist entirely of integers. In addition, we compute and compare the associated distance energies, namely, the distance energy ED, the distance Laplacian energy EDL, and the distance signless Laplacian energy EDQ to explore their structural significance. Using computational tools, we present numerical results and graphical comparisons that reveal meaningful relationships among these energies. In particular, our analysis establishes the conjecture in the form of a strict inequality EDL>EDQ>ED. These findings demonstrate that the distance Laplacian energy is more sensitive to molecular structural variations, highlighting its effectiveness as a discriminative molecular descriptor in chemical graph theory.

The concept of metric dimension plays a vital role in numerous fields such as computer science, image processing, pattern recognition, integer programming, pharmaceutical research, and chemical production. The metric dimension of a graph represents the smallest set of vertices capable of uniquely identifying all other vertices based on their distances. However, determining the local fractional metric dimension (LFMD) remains a challenging task due to its computational complexity. In this study, we explore the LFMD of various subclasses of [Formula: see text] graphs, including fan, path, Toeplitz, cycle, and zero-divisor graphs. We derive and analyze the upper bounds of these structures, establishing general patterns for their LFMDs. Our results reveal that the upper bound for [Formula: see text] graphs is [Formula: see text]; for [Formula: see text] it is 2; for [Formula: see text] it is [Formula: see text]; for [Formula: see text] it is 1 when [Formula: see text] and [Formula: see text] is even, and [Formula: see text] is [Formula: see text] when [Formula: see text] and [Formula: see text] is odd; otherwise, it is always [Formula: see text]; and for [Formula: see text] it is [Formula: see text]. These findings contribute to a deeper understanding of the structural properties of [Formula: see text] graphs and open new directions for their applications in complex network modeling and computational analysis.

A bijection for descent sets of permutations with only even and only odd cycles

The diurnal anisotropy of galactic cosmic ray intensity provides an important diagnostic of solar modulation processes operating in the heliosphere. In the present work, a comprehensive long-term investigation of the diurnal anisotropy has been carried out using pressure-corrected hourly neutron monitor data from two high-latitude stations, Moscow and Kiel, covering a period of 52 years (1965–2016) corresponding to solar cycles 20 to 24. Daily values of the first harmonic diurnal amplitude and phase were obtained through harmonic analysis, from which annual mean values were derived. The long-term relationship between the diurnal amplitude and phase has been examined on annual, solar-cycle, and solar magnetic polarity bases. While day-to-day variations of amplitude and phase are found to be statistically independent, their annual averages exhibit clear solar-cycle-dependent behavior. A weak positive correlation between the diurnal amplitude and phase is observed when all years are considered together. However, a pronounced odd–even solar cycle asymmetry emerges when the data are segregated into odd and even cycles. Strong and statistically significant positive correlations characterize the odd solar cycles, whereas even solar cycles display weak and insignificant correlations. Furthermore, a clear dependence on solar magnetic polarity is observed, with large positive correlations during positive polarity epochs (A > 0) and reduced or reversed correlations during negative polarity epochs (A < 0). These results provide strong observational evidence that long-term variations of cosmic ray diurnal anisotropy are governed not only by solar activity but also by heliospheric magnetic polarity and particle drift effects.

The rose graph, denoted by R(C_n), n>=3 constructed by a cycle graph C_n with n isolated vertices that connect every two vertices in the cycle graph with one isolated vertex. The rose barbell graph, denoted by B_{R(C_n)} is a simple graph formed by connecting two rose graphs R(C_n) by edges v_1,v^{'}_1 as a bridge. In this paper, we determined the partition dimension of the rose graph and its barbell

Within computing and information technology, interconnection systems are often modeled using graphs. A notable example is the swapped Optical Transpose Interconnection System (OTIS). Ensuring resilience to faults is crucial in optoelectronic systems. Across different fault classifications that can arise in an interconnection structure, two notable ones are the failure of a node (such as a processor in the case of OTIS over base graph G, denoted by O G and the disruption in facilitating communication between nodes breakdown of processor-to-processor communication). To manage these faults, it is crucial to assign a unique identifier to each node. In terms of graph theory, this corresponds to identifying the metric dimension, denoted as β ( G ) , and Fault-oriented metric dimension β ′ ( G ) of the graph G representing the interlinking structure. This paper analyzes the OTIS architecture based on C m (Cycle graph containing m vertices), represented as O C m , for resolvability and resolvability under fault conditions. It is proved that β ( O C m ) = m − 1 and β ′ ( O C m ) = m .

Self-testing identifies quantum states and correlations that exhibit nonlocality, distinguishing them, up to local transformations, from other quantum states. Due to their strong nonlocality, it is known that all graph states can be self-tested in the standard setting – where parties are not allowed to communicate. Recently it has been shown that graph states display nonlocal correlations even when bounded classical communication on the underlying graph is permitted, a feature that has found applications in proving a circuit-depth separation between classical and quantum computing. In this work, we develop self testing in the framework of bounded classical communication, and we show that certain graph states can be robustly self-tested even allowing for communication. In particular, we provide an explicit self-test for the circular graph state and the honeycomb cluster state – the latter known to be a universal resource for measurement based quantum computation. Since communication generally obstructs self-testing of graph states, we further provide a procedure to robustly self-test any graph state from larger ones that exhibit nonlocal correlations in the communication scenario.

Spectral extremal problems for non-bipartite graphs without odd cycles

Probability distributions are essential tools for modeling, prediction, and statistical inference. In recent years, several generalized families of distributions have been proposed to extend classical models and increase their flexibility in capturing complex data behaviors. This paper reviews selected generalized families published between 2023 and 2025, focusing on their construction mechanisms, statistical properties, estimation methods, and real-world applications. The families discussed include trigonometric-based, inverse, Lomax-generated, Topp–Leone, and hybrid forms. To illustrate their performance, five families were combined with the exponential distribution and fitted to a real dataset. The comparison shows that all extended models provide an adequate fit, while the standard exponential model performs poorly. The findings confirm the practical value of generalized families in improving data modeling.Let η ≥ 3 be an integer with primitive root π. For a simple connected graph G of order n, a bijective function f : V (G) → {1, 2, …, n} is called a logarithmic cordial labeling to the base π modulo η if the induced function f ∗ π,η : E(G) → {0, 1}, defined by f ∗ π,η(ab) = 0 if indπ,η(f(a) + f(b)) ≡ 0(mod 2) or gcd(f(a) + f(b), η) ̸= 1, and f ∗ π,η(ab) = 1 if indπ,η(f(a) + f(b)) ≡ 1(mod 2), satisfies the condition |ef∗π,η (0) − ef∗π,η (1)| ≤ 1 where ef∗π,η (i) is the number of edges with label i(i = 0, 1). In this paper, we study the logarithmic cordial labeling of various classes of graphs, including path graphs, cycle graphs, star graphs, and complete graphs.

Let η ≥ 3 be an integer with primitive root π. For a simple connected graph G of order n, a bijective function f : V (G) → {1, 2, …, n} is called a logarithmic cordial labeling to the base π modulo η if the induced function f ∗ π,η : E(G) → {0, 1}, defined by f ∗ π,η(ab) = 0 if indπ,η(f(a) + f(b)) ≡ 0(mod 2) or gcd(f(a) + f(b), η) ̸= 1, and f ∗ π,η(ab) = 1 if indπ,η(f(a) + f(b)) ≡ 1(mod 2), satisfies the condition |ef∗π,η (0) − ef∗π,η (1)| ≤ 1 where ef∗π,η (i) is the number of edges with label i(i = 0, 1). In this paper, we study the logarithmic cordial labeling of various classes of graphs, including path graphs, cycle graphs, star graphs, and complete graphs.

This study presents a detailed exploration of the Revan family of topological indices and their polynomial representations for diverse classes of quadrilateral snake graphs. Originating from the concept introduced by V. R. Kulli, Revan indices integrate both the minimum and maximum vertex degrees, which provide a refined measure of graph connectivity and structure. The research systematically derives explicit analytical expressions for Revan indices and their corresponding polynomials for four principal graph variants: standard, alternate, double, and cyclic quadrilateral snake graphs, highlighting their distinctive structural characteristics and degree-based relationships. To complement the theoretical formulations, a Python-based computational framework was developed to automate the calculation and symbolic representation of these indices. This implementation enables efficient validation of analytical results and facilitates the extension of Revan-based metrics to larger and more complex graph families. The findings underscore the potential of Revan indices as powerful structural descriptors in mathematical chemistry and network theory, with promising applications in quantitative modeling, cheminformatics, and the broader field of graph-based molecular design.