On the coefficients of the Laplacian characteristic polynomial of trees


I. Gutman, Ljiljana PavloviŠ


Let the Laplacian characteristic polynomial of an $n$-vertex tree $T$ be of the form $\psi(T,\lambda) = \sum\limits_{k=0}^n (-1)^{n-k}\,c_k(T)\,\lambda^k$ . Then, as well known, $c_0(T)=0$ and $c_1(T)=n$ . If $T$ differs from the star ($S_n$) and the path ($P_n$), which requires $n \geq 5$ , then $c_2(S_n) < c_2(T) < c_2(P_n)$ and $c_3(S_n) < c_3(T) < c_3(P_n)$ . If $n=4$ , then $c_3(S_n)=c_3(P_n)$ .