On the number of spanning trees and asymptotic behavior of certain classes of sprocket wheel graphs

Ahmad Asiri; Salama Nagy Daoud; Ahmad Asiri; Salama Nagy Daoud

doi:10.3934/math.2026791

AIMS Mathematics

2026, Volume 11, Issue 7: 19457-19487. doi: 10.3934/math.2026791

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On the number of spanning trees and asymptotic behavior of certain classes of sprocket wheel graphs

Ahmad Asiri ¹,
Salama Nagy Daoud ^{2,3
,
,}

1.
Department of Mathematics, Applied College at Mahail Aseer, King Khalid University, Abha 61421, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia
3.
Department of Mathematics and Computer Sciences, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt

Received: 21 February 2026 Revised: 17 May 2026 Accepted: 27 May 2026 Published: 02 July 2026
MSC : 05C05, 05C50, 05C99

The sprocket wheels are specialized, toothed wheels that transfer rotating motion by engaging with chains, belts, or perforated materials. They are frequently seen on equipment, motorcycles, and bicycles. They do not mesh directly with one another like gears do. They are characterized by their teeth count and pitch and are usually composed of metal or sturdy polymers, guaranteeing accurate, long-lasting, and dependable power transfer. In this study, we used block matrices, linear algebraic techniques, and the properties of Chebyshev polynomials to obtain explicit formulas for the number of spanning trees of new classes of sprocket wheel graph families. Additionally, we studied the asymptotic entropy of spanning trees on our graphs. Last, we compared the entropies of our graphs to those of other studied graphs with average degreed of 4 and 5.
- number of spanning trees,
- sprocket wheel graph,
- Chebyshev polynomials,
- entropy
Citation: Ahmad Asiri, Salama Nagy Daoud. On the number of spanning trees and asymptotic behavior of certain classes of sprocket wheel graphs[J]. AIMS Mathematics, 2026, 11(7): 19457-19487. doi: 10.3934/math.2026791

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Abstract

The sprocket wheels are specialized, toothed wheels that transfer rotating motion by engaging with chains, belts, or perforated materials. They are frequently seen on equipment, motorcycles, and bicycles. They do not mesh directly with one another like gears do. They are characterized by their teeth count and pitch and are usually composed of metal or sturdy polymers, guaranteeing accurate, long-lasting, and dependable power transfer. In this study, we used block matrices, linear algebraic techniques, and the properties of Chebyshev polynomials to obtain explicit formulas for the number of spanning trees of new classes of sprocket wheel graph families. Additionally, we studied the asymptotic entropy of spanning trees on our graphs. Last, we compared the entropies of our graphs to those of other studied graphs with average degreed of 4 and 5.

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