On the Matching Polynomials of the Complete \(n\)-Partite Graph

Let \(G\) be a simple graph with \(n\) vertices. \(p(G,k)\) denotes the number of ways in which one can select \(k\) independent edges in \(G\) (\(k \geq 1\)). Let \(p(G,0) = 1\) for all \(G\).
The matching polynomial \(\alpha(G)\) of a graph \(G\) is given by:
\[\alpha(G) = \alpha(G,x) = \sum_{k=0}^{\left[\frac{n}{2}\right]} (-1)^k p(G,k) x^{n-2k}\]

In this article, we give the matching polynomials of the complete \(n\)-partite graph with a differential operator.