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Related Concept Videos

Entropy02:39

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
Entropy01:18

Entropy

The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Absolute Entropies and the Third Law of Thermodynamics01:23

Absolute Entropies and the Third Law of Thermodynamics

Ludwig Edward Boltzmann developed a definition for entropy, which stated that absolute entropy is proportional to the natural logarithm of the number of possible combinations of particles. Entropy stands alone among state functions as the only one whose absolute values can be determined.Consider a gas sample confined to a container. As the container expands, the energy levels of gas molecules become more closely spaced. This increases the number of available energy states, thereby increasing...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

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Related Experiment Video

Updated: May 22, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Prodeg type Shannon graph entropies with closed forms bounds and QSPR modeling.

Mohammed Alsharafi1,2, Yusuf Zeren3,4

  • 1Department of Mechatronics Engineering, Faculty of Engineering and Computing, University of Science and Technology, Aden, Yemen. alsharafi205010@gmail.com.

Scientific Reports
|May 20, 2026
PubMed
Summary

This study introduces Prodeg entropies, a new class of graph invariants for quantifying molecular complexity. These novel graph entropies show strong correlations with established molecular descriptors and improve quantitative structure-activity relationship models in chemistry.

Keywords:
Chemical graph theoryDegree-based graph entropyMolecular descriptorsNordhaus–Gaddum inequalitiesProdeg indicesQSPR/QSARShannon entropyTensor product of graphs

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Related Experiment Videos

Last Updated: May 22, 2026

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Area of Science:

  • Graph theory
  • Mathematical chemistry
  • Network analysis

Background:

  • Degree-based graph entropies quantify structural heterogeneity using Shannon entropy.
  • Existing methods lack a unified framework for certain degree-power invariants.

Purpose of the Study:

  • Develop a unified framework for Prodeg-type degree-power invariants and their associated entropies.
  • Derive closed-form expressions and establish extremal behaviors for various graph families.
  • Investigate the chemical relevance and QSPR modeling capabilities of these new entropies.

Main Methods:

  • Developed a general framework for degree-weighted distributions and Shannon entropy.
  • Derived closed-form expressions for complete graphs, cycles, paths, stars, and complete bipartite graphs.
  • Utilized tensor-product principles and majorization for theoretical analysis.
  • Analyzed chemical compounds from the ChEMBL database and benchmarked QSPR models.

Main Results:

  • Established a unified framework for Inverse Prodeg, Misbalance Prodeg, and Yemen Prodeg entropies.
  • Derived sharp extremal behaviors and bounds for connected graphs.
  • Demonstrated that Prodeg entropies strongly correlate with established molecular complexity measures (BertzCT, AvgIpc).
  • Achieved competitive performance in QSPR modeling, with tree ensembles showing high accuracy for size-related properties.

Conclusions:

  • Prodeg entropies offer a compact and interpretable alternative to classical graph descriptors.
  • These entropies enhance QSPR models, particularly when combined with existing descriptors.
  • The unified framework provides a versatile tool for chemical graph analysis and molecular property prediction.