Kirchhoff Index Calculation Special Graphics

The resistance distance rijbetween two vertices viand vjof a (connected, molec-ular) graph G is equal to the efective resistance between the corresponding two pointsof the electrical network, constructed so as to correspond to G, such that the resistanceof any edge is unity. The Kirchhof index Kf(G), defined by Klein et al in1993, isthe sum of resistance distance between all pairs of vertices in G which lately shown tobe a topological invariant of the molecular graph G modeled. In this thesis, using theautomorphism group and the Laplacian spectrum of the graph, we discuss the Kirchhofindices of two classes of graphs, which are linear pentagonal chains and Cyclopolyacenes.This thesis contains four parts. In the first part, we give some definitions andlemmas about the Kirchhof index of graphs. In the second part, according to theautomorphism group and the decomposition theorem of Laplacian polynomial, we obtainthe block matrix of the Laplacian matrix. The third and fourth parts include the mainresults of this thesis.