Resistance Based On Distance And Distance Chart Resistance Graph Invariants Research

The resistance distance Ω_G(i, j) between vertices i and j of a connected graph G iscomputed as the efective resistance between nodes i and j in the corresponding networkconstructed from G by replacing each edge of G with a unit resistor. The global cyclicityindex of G is defined as the sum of conductance excess between all pairs of adjacentvertices.In this thesis, we mainly study resistance distances and the global cyclicity indexof graphs. The thesis consists of three chapters. In chapter1, we first introduce somebasic concepts, terminology and notations. Then we point out the physical-chemicalresearch backgrounds. Finally, we survey the developments in the study of resistancedistances and the global cyclicity index.In the second chapter, by electrical network theoretical laws such as Rayleigh’sshort-cut principle and mesh current method, we establish a lower bound on resistancedistances via vertex degrees and characterize necessary and sufcient conditions underwhich the lower bound is attained.In the last chapter, by utilizing the number of spanning trees and2-spanning forests,we obtain resistance distances between pairs of adjacent vertices in linear hexagonalchains. Consequently, the global cyclicity index of hexagonal chains is derived in termsof resistance distances.