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The mysterious connection between molecular structure and physical properties: How does the Wiener index affect boiling point?
In chemical graph theory, the Wiener index is a molecular topological index proposed by Harry Wiener in 1947, which represents the sum of the shortest path lengths between all non-hydrogen atoms in the molecule. This index is not only of great significance in chemical molecular structure, but also plays a key role in areas such as computer networks and strengthening the security of lattice hardware.
The Wiener index is the oldest topological index related to molecular branching.
The Wiener index was originally called "path number", and based on its success, many chemical graph topological indices based on information from distance matrices emerged. This quantity is also studied in mathematics under different names, including "transmission of graphs", etc. In social network theory, the Wiener index is also closely related to the closeness centrality of a vertex in a graph. This index is inversely proportional to the sum of the distance of that vertex to all other vertices and is often used in sociological measurements.
Examples of Wiener index
Take butane (C4H10) as an example. It has two different structural isomers: n-butane (n-butane) and isomers. Butane (isobutane). The chemical diagram of n-butane is a path diagram with four vertices, while the chemical diagram of isobutane is a tree with three leaves connected by a central vertex.
The structural differences between n-butane and isobutane make their Wiener indices significantly different.
The calculation results show that the Wiener index of n-butane is 10, while the Wiener index of isobutane is 9. Although the two molecules have the same chemical formula and the same number of carbon-carbon and carbon-hydrogen bonds, their different structures result in different Wiener indices.
Relationship between Wiener index and chemical properties
Early research pointed out that the Wiener index is closely related to the boiling point of alkane molecules. Subsequent studies on quantitative structure-activity relationships found that it is also related to other properties, including critical point parameters, liquid phase density, surface tension and viscosity.
The Wiener index can help us predict the physical properties of molecules.
For example, as the Wiener index of an alkane molecule increases, its boiling point tends to increase. This phenomenon showed how changes in molecular structure affect the physical properties of molecules and sparked further interest in molecular design and chemical synthesis.
Calculation method of Wiener index
The Wiener index can be used to calculate the distance between all pairs of vertices using an algorithm. In an unweighted graph, these distances can be computed by repeatedly performing a breadth-first search. For weighted graphs, the Floyd-Warshall algorithm or Johnson algorithm can be used for calculation.
The application of these algorithms greatly improves the efficiency of Wiener index calculations.
In certain types of graphs, such as tree structures, the Wiener exponent can be calculated more efficiently. For example, dividing a tree by removing a single edge, calculating the sum of the Wiener exponents of the two subtrees, and adding the length of the path through that edge is called the "divide and conquer algorithm."
Exploration of inverse problems
In 1995, Gutman and Yeh considered the question of which numbers could be represented as the Wiener exponents of graphs. They found that with two exceptions, all positive integers could be represented. This result reveals the diversity and complexity of Wiener indices and inspires further research.
These findings remind us that there are still endless possibilities for exploration in graph theory.
Conclusion
The Wiener index is not just a number. What lies behind it is the profound connection between the molecular structure and its physical properties. Especially in the research of chemistry and materials science, its importance is self-evident. Future research may reveal more mysterious connections between such indices and physical properties. Will this affect our understanding and application of molecular design?
