Counting spanning trees in generalized $K_n$-chain/ring graphs

Let $K_n$ be the complete graph of order $n$. Very recently, the number of spanning trees (the $\mathcal{N}\mathcal{S}\mathcal{T}$ for short) and the resistance distances in $K_n$-chain (ring) graphs were determined explicitly. We generalized the concept to the generalized $K_n$-chain (ring) graph $[L_n^s{]}^m$ ( $[C_n^s{]}^m$). New formulae for the $\mathcal{N}\mathcal{S}\mathcal{T}$ of $[L_n^s{]}^m$ and $[C_n^s{]}^m$ were given by a simple and more physical way with a novel technique of adding a pair of positive and negative edges, avoiding complicated linear algebraic computations.