Combinatorics of random walks on graphs and walk-entropies: generalized Petersen and isomerization graphs

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2025-03-15 DOI:10.1007/s10910-025-01712-1

Krishnan Balasubramanian

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引用次数: 0

Abstract

We consider the combinatorial enumeration of random walks on graphs with emphasis on symmetric, vertex-transitive and bipartite generalized Petersen graphs containing up to 720 vertices. We enumerate self-returning and non-returning walks originating from each vertex of graphs using the matrix power algorithms. We formulate the vertex entropies, scaled unit self-return and non-return walk entropies of structures which provide measures for the combinatorial complexity of graphs. We have chosen mathematically and chemically interesting generalized Petersen graphs G(n,k) with floral symmetries, as they find several applications in dynamic stereochemistry and several other fields. These studies reveal several interesting walk patterns and walk sequences for these graphs, and paves the way for statistical studies on these chemically and mathematically interesting graphs. Moreover, walk-based vertex partitions are machine-generated from the enumerated walk n-tuple vectors, although they do not always correlate with the automorphic partitions. Hence the present study attempts to integrate statistical mechanics, graph theory, combinatorial complexity, and symmetry for large molecular and biological networks.

Graphical abstract

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图上随机游走与游走熵的组合：广义Petersen图与异构化图

我们考虑图上随机游走的组合枚举，重点考虑包含720个顶点的对称、顶点传递和二部广义Petersen图。我们使用矩阵幂算法枚举从图的每个顶点出发的自返回和不返回的行走。给出了结构的顶点熵、标度单位自返回和不返回行走熵，为图的组合复杂度提供了度量。我们选择了数学上和化学上有趣的广义彼得森图G（n,k）与花对称，因为它们在动态立体化学和其他几个领域有很多应用。这些研究揭示了这些图的一些有趣的行走模式和行走序列，并为这些化学和数学上有趣的图的统计研究铺平了道路。此外，基于行走的顶点分区是机器从枚举的行走n元组向量生成的，尽管它们并不总是与自同构分区相关。因此，本研究试图将统计力学、图论、组合复杂性和大型分子和生物网络的对称性结合起来。图形抽象

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来源期刊

Journal of Mathematical Chemistry 化学-化学综合

CiteScore

3.70

自引率

17.60%

发文量

105

审稿时长

6 months

期刊介绍： The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches. Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.