The number of spanning trees for Sierpiński graphs and data center networks

Abstract

The number of spanning trees is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. In 2007, Chang et al. proposed two conjectures on the number of spanning trees of Sierpiński triangle graphs and its spanning tree entropy. In this paper, we completely confirm these conjectures. For data center networks $D_{k,n}$, we get the exact formula for $k=1$, and upper and lower bounds for $k\ge 2$. Our results allow also the calculation of the spanning tree entropy of Sierpiński graphs and data center networks.