Path energy of two kinds of special tricyclic graphs

Abstract

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.

Keywords

• graph theory; real symmetric matrix; eigenvalue; tricyclic graph; path matrix; path energy