On Laplacian Characteristic Polynomials of Trees

Bo Zhou
Department of Mathematics, South China Normal University, Guangzhou 510631, P. R. China

Abstract     Full Text  PDF


Let $G$ be a graph on $n$ vertices and let $L(G)$ be the Laplacian matrix of $G$. The Laplacian characteristic polynomial of $G$ is defined as $\psi
(G, x)=\det(xI_n-L(G))$. We write it as $ \psi(G,
x)=\sum_{k=0}^n (-1)^{k}c_{k}(G)x^{n-k}$. Let $T$ be a tree on $n$ vertices and let $k$ be an integer, $2\le k\le n-2$. Then $c_k(T)>c_k(S_n)$ if $T\not=S_n$ and $ c_k(T)<c_k(P_n)$ if $T\not=P_n$, where $S_n$ and $P_n$ denote the star and path on $n$ vertices, respectively.