The Laplacian characteristic polynomial of the path-tree of the complete graph

Let $T_n$ be the path tree of the complete graph $K_n$. Denote by $H_n$ the weighted path with vertex set $V\left(H_n\right)=\left\{v_i\right|1\le i\le 2n-1\}$, edge set $E\left(H_n\right)=\left\{v_i{v}_{i+1}\right|1\le i\le 2n-2\}$, weighted function $w\left(v_i{v}_{i+1}\right)=i/2$ if $i$ is even and $w\left(v_i{v}_{i+1}\right)=1$ otherwise. Guo and Chen (2024) proved that the matching polynomial of $T_n$ can be represented by the matching polynomials of the compete graphs. In this paper, we show that the Laplacian characteristic polynomial $\sigma (T_n,x)$ of $T_n$ satisfies: $\sigma (T_n,x^2)=x{a}_n\left(x\right)\prod _{i=2}^n{b}_i{\left(x\right)}^{{(n-1)}_{n-i+2}/(i-1)},$where $a_n\left(x\right)$ and $b_i\left(x\right)$ are the matching polynomials of $H_n$ and $H_i-v_{2i-1}$, respectively, and ${(n-1)}_{n-i+2}=(n-1)(n-2)\cdots (i-2)$.