TITLE: Number of Spanning Trees in a certain graph
QUESTION [5 upvotes]: Let $k,n\in\mathbb N$ and define the simple graph $G_{k,n}=([n],E)$, where $ij\in E\Leftrightarrow 0 <|i-j|\leq k$ for $i\neq j\in [n]$.
I need to calculate the number of different spanning trees.
I am applying Kirchoff's Matrix Tree theorem to solve this but i am not getting the answer. for example : $k=3$ and  $n=5$, my matrix is 
\begin{pmatrix}
3 & -1 &-1& -1&  0\\
-1 & 4 &-1 &-1 &-1\\
-1 &-1 & 4 &-1 &-1\\
-1 &-1 &-1 & 4 &-1\\
0 & -1 &-1 &-1  &3
\end{pmatrix}
and the final answer  as per the Kirchoff's theorem is the determinant of any of the co-factor of the matrix. proceeding in the same way i am getting something else but the answer is 75. 
Is there any another approach to solve this problem or my process is wrong? please help
Thank you

REPLY [2 votes]: The answer seems correct. You can check with a different method in this case, because the graph you are considering is the complete graph minus one specific edge E.
By Cayley's formula, there are $5^3=125$ spanning trees of the complete graph on 5 vertices. Each such tree has four edges, and there are 10 possible edges in the complete graph. By taking a sum over all edges in all spanning trees, you can show that $\frac{2}{5}$ of the spanning trees will contain the specific edge $E$. So the remaining number of spanning trees is $\frac{3}{5} \times 125 = 75$, which agrees with your answer.