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 $K_{m,n}$

TL;DR

This paper introduces a new method based on vertex deletion to compute the number of spanning trees for generalized cones of complete graphs and certain modified bipartite graphs, providing explicit formulas and extending existing techniques.

Contribution

It presents a novel vertex deletion approach for counting spanning trees in generalized cones and modified bipartite graphs, expanding the toolkit for graph enumeration.

Findings

Derived formulas for spanning trees of generalized cones of $K_n$

Extended methods to modified bipartite graphs $K_{m,n}$

Introduced a new vertex deletion-based technique

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.