The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices

Abstract

The complexity (number of spanning trees) in a finite graph Γ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning trees. In graph theory, one constantly strives to create novel structures from existing ones. The super subdivision operation produces more complicated networks, and the matrices of these networks can be divided into block matrices. Using methods from linear algebra and the characteristics of block matrices, we derive explicit formulas for determining the complexity of the super subdivision of a certain family of graphs, including the cycle Cn, where n=3,4,5,6; the dumbbell graph Dbm,n; the dragon graph Pm(Cn); the prism graph Πn, where n=3,4; the cycle Cn with a Pn2-chord, where n=4,6; and the complete graph K4. Additionally, 3D plots that were created using our results serve as illustrations.