On the Characteristic Polynomial of a Hexagonal System and Its Application

DOI：10.3770/j.issn:2095-2651.2014.03.002

 作者 单位 娄贞贞 新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046 黄琼湘 新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046

$L_n$表示直链六角系统图, $F_n$表示环链六角系统图, $M_n$表示M\"{o}bius环链六角系统图. 1968年, Derflinger和Sofer用群论的方法给出了$L_n$和$F_n$的特征多项式. 之后, Gutman通过使用Godsil和McKay的一个多项式的结果给出了这两类图的谱. 本文采用一种直接的办法给出了$F_n$和$L_n$的特征多项式和谱,并用这个办法给出了$M_n$的特征多项式和谱,这是个新结果. 另外,我们还得到了这三类图的零度和能量,考虑了$L_n$和 $M_n$的界.

Let $L_n$ be the hexagonal chain graph，$F_n$ be the hexacyclic system graph and $M_n$ be the M\"{o}bius hexacyclic system graph. Derflinger and Sofer gave the of $L_n$ and $F_n$ by using group theoretical method. Later, Gutman gave the spectra of them using a polynomial result due to Godsil and McKay. In this paper, we givespectra a simple and direct method to determine the characteristic polynomial and spectra of $F_n$ and $L_n$. By the method, we give the characteristic polynomial and spectrum of $M_n$ that is new. Additionally, the exact values of total $\pi$-electron energy and the nullities of $L_n$, $F_n$ and $M_n$ are obtained, and the bounds for the energy of $L_n$ and $M_n$ are also considered.