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

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.