On the coefficients of the Laplacian characteristic polynomial of trees

I. Gutman and Ljiljana Pavlovic

Abstract: 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 $c_3(S_n)=c_3(P_n)$\,.

Keywords: Laplacian spectrum, Laplacian characteristic polynomial, Trees, Distance (in graph), Wiener number

Classification (MSC2000): 05C05, 05C12, 05C50

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