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We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic graphs. Then we show how it can be efficiently computed for large classes of chemically interesting graphs using a variant of the cut method introduced by Klavžar, Gutman and Mohar. Explicit formulas are presented for several classes of benzenoid graphs and Cartesian products. At the end we state several conjectures and list some open problems.

Spiking neural networks can exhibit complex emergent dynamics, but the credibility of spatially explicit agent-based implementations depends on systematic verification and validation (V&V). This study introduces LANA (Local Adaptive Neural Agents), an agent-based spiking neural network in which neurons, propagating signals, directed synapses, and a diffusive environmental field are represented as distinct interacting components. We present a five-level V&V framework spanning operator-level tests, single-neuron mechanisms, propagation behavior, network-level dynamics, and sensitivity/robustness analysis. Across 13 predefined tests and approximately 2000 simulation runs, the model satisfied all prespecified pass criteria: synaptic delays reproduced the expected propagation law exactly, environmental decay and diffusion matched analytical expectations, threshold and refractory mechanisms behaved as predicted, inhibition suppressed firing monotonically, and environmental coupling induced a transition toward higher variability and oscillatory-like activity. Matched-seed comparisons further showed that explicit signal transport and environmental feedback substantially amplify activity relative to a neuron-only baseline while leaving synaptic delay propagation unchanged. Additional regime and lesion experiments demonstrated distinct resting, hyperexcitable, and focal-lesion states, with the lesion condition producing an acute decline followed by only partial recovery. Together, these results provide a transparent V&V baseline for LANA and illustrate how agent-based spiking models can be tested and interpreted across multiple scales.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by α, these operators offer a unique flexibility: setting α=1 recovers the classical inequalities for harmonically convex functions, while setting α=0 yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators.

Agent-based neural models often encode transmission within neuron state updates, which can make it difficult to separately log and quantify spatial recruitment patterns, delay structure, and environment-mediated feedback effects. We present LANA (Local Adaptive Neural Agents), a dual-agent neural agent-based model in which neurons and propagating signals are represented as distinct interacting entities embedded in a dynamic environmental field. The model combines discrete leaky integrate-and-fire neuron dynamics, mobile signal agents, synaptic links with distance-dependent delays, and a bounded environment-to-neuron feedback mechanism. LANA is intended as a normalized phenomenological mesoscopic framework for mechanism-level comparison rather than as a circuit-specific biophysical reconstruction. To support interpretability and reproducibility, we report a compact internal verification block for the implemented operators, including delay propagation, environmental decay and diffusion, threshold activation, and refractory enforcement. We then compare the full LANA model against a matched neuron-only baseline and summarize spatial recruitment using first-spike maps, cumulative recruitment times, and wavefront speed as a secondary descriptive metric. Finally, we evaluate two controlled operating regimes, a resting regime (S1) and a hyperexcitable regime (S2), under fixed network size, stimulation schedule, and matched random seeds. Relative to the baseline, the full model sustains and spreads activity more effectively and provides spatially resolved recruitment summaries, including first-spike timing and cumulative recruitment measures, that are not available in the same form when transmission is represented only through neuron-level updates. Relative to S1, S2 exhibits earlier activation, higher firing activity, stronger environmental accumulation, and faster cumulative recruitment. Local and factorial sensitivity analyses further identify the parameters that most strongly govern these regime differences. Together, these results position LANA as a normalized mesoscopic and computationally tractable framework for studying how excitability, transport state dynamics, delayed coupling, and environment-mediated feedback jointly shape emergent activity in controlled simulation settings.

Fractional calculus has been a concept used to acquire new variants of some well-known integral inequalities. In this study, our primary goal is to develop majorized fractional Simpson’s type estimates by employing a differentiable function. Practicing majorization theory, we formulate a new auxiliary identity by utilizing fractional integral operators. In order to obtain new bounds, we employ the idea of convex functions on the Niezgoda–Jensen–Mercer inequality for majorized tuples, along with some fundamental inequalities including the Hölder, power mean, and Young inequalities. Some applications to the quadrature rule and examples for special functions are provided as well. Interestingly, the main findings are the generalizations of many known results in the existing literature.

The classical Grüss and related inequalities have spurred a range of improvements, refinements, generalizations, and extensions. In the present article, we provide generalizations of Sokolov’s inequality in weighted Lebesgue LωΩ,A,μ spaces by employing the weighted Sonin’s identity and Čebyšev functional. As a result, we provide a generalized Grüss inequality in which the bounding constants are improved with bounding functions in LωpΩ,A,μ spaces. As an application, we provide several new bounds for Jensen–Grüss type differences.

Over the past three decades, fractional calculus has gained increasing importance and practical relevance in various fields of science and engineering. This article aims to develop enhanced estimations based on the fractional Simpson’s rule for functions that are twice differentiable. Leveraging majorization theory, we introduce a novel auxiliary identity by making use of fractional integral operators. To derive the novel bounds presented in this manuscript, we employ the notion of convex functions in conjunction with the Niezgoda Jensen Mercer (JM) inequality for majorized tuples, as well as some core inequalities, including Young’s, Power mean, and Hölder’s inequalities. Furthermore, this study encompasses the application of quadrature rules and provides illustrative examples related to special functions. Notably, the primary contributions of this research involve the extension and generalization of numerous wellestablished findings found in the current body of literature.

This paper focuses on introducing and investigating the class of generalized$ n $ -fractional polynomial$ s $ -type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowski-type inequalities are presented for generalized$ n $ -fractional polynomial$ s $ -type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized$ n $ -fractional polynomial$ s $ -type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the Riemann-Liouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.