Topological irregularity and entropy signatures of generalized silicate cage networks via generalized $M$-polynomials

Duyu Zhang; Xiaohong Dong; Hongjie Zhang; Haoyu Kong; Duyu Zhang; Xiaohong Dong; Hongjie Zhang; Haoyu Kong

doi:10.3934/math.20261280

AIMS Mathematics

2026, Volume 11, Issue 10: 32538-32571. doi: 10.3934/math.20261280

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Research article

Topological irregularity and entropy signatures of generalized silicate cage networks via generalized $M$-polynomials

1.
School of Mathematics, Hefei University of Technology, Hefei 230601, China
2.
College of Civil Engineering, Anhui Jianzhu University, Hefei 230601, China
3.
School of Electronics and Information Engineering, Anhui Jianzhu University, Hefei 230601, China
4.
Faculty of Science, The University of Melbourne, Melbourne 3000, Australia

Received: 28 July 2026 Revised: 04 September 2026 Accepted: 16 September 2026 Published: 09 October 2026
MSC : 05C09, 05C10, 05C92

Silicate cage frameworks, assembled from interconnected Si–O units, combine chemically relevant porosity and connectivity with structural motifs that are naturally amenable to graph-theoretical characterization. First, the parity-dependent structures of the generalized silicate cage network $SLC_{\mu}(\nu)$ were distinguished through degree partitions and generalized $M$-polynomials for even- and odd-layer configurations. The inherited partitions were checked by both edge totals and degree sums, and the degree-$5$ boundary corrections of the odd case were made explicit. Second, differential and spectral operators yielded exact closed-form expressions for twelve degree-based irregularity indices from the common $M$-polynomial representation. Third, exact parameter-sensitivity relations quantified the parity-preserving layer increments, the proportional effect of replication, and the common quadratic leading behavior under layer expansion. Representative exact surfaces illustrated distinct absolute, squared, normalized, and logarithmic weighting mechanisms without redundant graphical displays. Finally, descriptor-weighted Shannon entropy formulas were established for both parity classes; positive-scale invariance reduced the twelve descriptor entropies to eleven distinct expressions, while the new scaling law gave logarithmic replication growth and universal large-layer behavior. These results retained the exact structural information of both parity classes and isolated how local degree heterogeneity, cage replication, and layer number jointly organize the network, condensing the numerical presentation while preserving the mathematical distinctions needed for interpreting parity-dependent structural complexity. The resulting framework supplies exact structural fingerprints for comparison and classification, as well as candidate structural variables for future property-oriented modeling when verified physicochemical data become available.
- generalized $M$-polynomial,
- silicate cage network,
- topological descriptor,
- irregularity index,
- descriptor-weighted Shannon entropy
Citation: Duyu Zhang, Xiaohong Dong, Hongjie Zhang, Haoyu Kong. Topological irregularity and entropy signatures of generalized silicate cage networks via generalized $M$-polynomials[J]. AIMS Mathematics, 2026, 11(10): 32538-32571. doi: 10.3934/math.20261280

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Abstract

Silicate cage frameworks, assembled from interconnected Si–O units, combine chemically relevant porosity and connectivity with structural motifs that are naturally amenable to graph-theoretical characterization. First, the parity-dependent structures of the generalized silicate cage network $SLC_{\mu}(\nu)$ were distinguished through degree partitions and generalized $M$-polynomials for even- and odd-layer configurations. The inherited partitions were checked by both edge totals and degree sums, and the degree-$5$ boundary corrections of the odd case were made explicit. Second, differential and spectral operators yielded exact closed-form expressions for twelve degree-based irregularity indices from the common $M$-polynomial representation. Third, exact parameter-sensitivity relations quantified the parity-preserving layer increments, the proportional effect of replication, and the common quadratic leading behavior under layer expansion. Representative exact surfaces illustrated distinct absolute, squared, normalized, and logarithmic weighting mechanisms without redundant graphical displays. Finally, descriptor-weighted Shannon entropy formulas were established for both parity classes; positive-scale invariance reduced the twelve descriptor entropies to eleven distinct expressions, while the new scaling law gave logarithmic replication growth and universal large-layer behavior. These results retained the exact structural information of both parity classes and isolated how local degree heterogeneity, cage replication, and layer number jointly organize the network, condensing the numerical presentation while preserving the mathematical distinctions needed for interpreting parity-dependent structural complexity. The resulting framework supplies exact structural fingerprints for comparison and classification, as well as candidate structural variables for future property-oriented modeling when verified physicochemical data become available.

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