On the Number of Spanning Trees of New Graph Families Created from the Star Graph and the Examination of Their Entropies

Complexity (number of spanning trees) is an essential and significant component in the design of communication networks (graphs). To ensure strong resistance and stiffness and to enhance the probability of a connection between two vertices, improvements to a network’s quality and perfection increase the number of trees that span it. Using block matrices and linear algebra techniques, we derive explicit formulas for the number of spanning trees of new graph families that are produced from star graphs in this study. The number of spanning trees in a graph is measured by the entropy of spanning trees, also known as asymptotic complexity, a graph theory metric that assesses the network’s structural robustness and dependability. Increased flexibility, stronger diverse connections, and improved resistance to random structural changes are all indicated by higher entropy. We also investigate the entropy of spanning trees on our graphs at the end of this study. Lastly, we compare the entropy of our graphs to that of other previously studied graphs with average degrees of four and five.