The Number of Spanning Trees for The Generalized Cones of $K_n$, The Generalized Half Cones of $K_{m,n}$ and Some Family of Modified $K_{m,n}$

Abstract

We compute the total number of spanning trees for the generalized cone of the complete graph $K_n$ and a number of families of some modified bipartite graphs $K_{m,n}$. In particular, we obtain a new method of finding the number of spanning trees of $K_n$ and $K_{m,n}$. Our method relies on the vertex deletion formula for the number of spanning trees.