2类特殊三圈图的路能量

针对三圈图种类较多且路矩阵复杂度较高的问题,运用矩阵分析方法、根的存在性定理及不等式的放缩,研究了2类三圈图有无悬挂点时的路能量。首先,分别给出2类三圈图有无悬挂点时的4种路矩阵,利用矩阵分析方法对实对称矩阵分块得出对应的特征多项式,由根的存在性定理及韦达定理判定出正负特征值的个数并估计出取值范围;其次,通过不等式的放缩求出2类三圈图有无悬挂点时的路能量。结果表明,2类三圈图在有无悬挂点时路矩阵负特征值的个数及取值范围是不一样的,对应的路能量也是不一样的。所得结果对后续三圈图的路能量极值问题研究具有一定的借鉴价值,也有利于推测相关化学分子结构的性质。

Path energy of two kinds of special tricyclic graphs

Aiming at the problem that there are many kinds of tricyclic graphs and the complexity of path matrix is high, the path energy of two kinds of tricyclic graphs with or without pendant vertices was studied by means of matrix analysis, the existence theorem of root and scaling of inequality. Firstly, four kinds of path matrices of two kinds of tricyclic graphs with or without pendant vertices were given, then the real-symmetric matrix was partitioned by matrix analysis to obtain the corresponding characteristic polynomials. The number of positive and negative eigenvalues was determined by the existence theorem of the root and the Vieta theorem, and the range of values was estimated. Secondly, the path energy of two kinds of tricyclic graphs with or without pendant vertices was found out by scaling inequality. The results show that the number and range of negative eigenvalues of the two kinds of tricyclic graphs are different when there are pendant vertices or not, so the corresponding path energy of tricyclic graphs is also different. The obtained results have certain reference value for the study of the path energy of extreme value problem of the tricyclic graphs, and it is conducive to speculate the structural properties of related chemical molecules.