Matching polynomials of path-trees of a complete bipartite graph

Abstract

Suppose that the vertex set of the complete bipartite graph $K_{s,t}$ is $X\cup Y$, where $\left|X\right|=s\le t=\left|Y\right|$. Let $T_{s,t}^w$ denote the path-tree of $K_{s,t}$ corresponding to vertex $w\in X\cup Y$. Then we show that the matching polynomial $\mu (T_{s,t}^w,x)$ is equal to $\mu (K_{s,t},x)\cdot$ $\left(\begin{array}{cc}{\left(\prod _{k=1}^{s-1}{\left(\mu {(K_{k,\alpha +k},x)}^{k-1}\mu {(K_{k,\alpha +k+1},x)}^{\frac{\alpha +k}{\alpha +k+1}}\right)}^{{\beta }_{s,t}^k}\right)}^t,& \text{if}w\in X;\\ \mu {(K_{s,t-1},x)}^{s-1}{\left(\prod _{k=1}^{s-1}{\left(\mu {(K_{k,\alpha +k-1},x)}^{(k-1)(\alpha +k)}\mu {(K_{k,\alpha +k},x)}^{\alpha +k-1}\right)}^{{\beta }_{s,t}^k}\right)}^s,& \text{if}w\in Y,\end{array}\right)$ where $\alpha =t-s,{\beta }_{s,t}^k={\prod }_{j=k+1}^{s-1}j(\alpha +j)$.

This result has a number of consequences. It follows that the roots of $\mu (T_{s,t}^w,x)$ (i.e. the eigenvalues of $T_{s,t}^w$), the (matching) energy of $T_{s,t}^w$, and the Hosoya index of $T_{s,t}^w$ can all be obtained directly from that of $K_{i,j}(1\le i\le s,1\le j\le t)$.