The Resistance Distances Between Two Vertices Of Some Network Graphs

Given an arbitrary graph,one can view each edge of the graph as an unit resistor,so that we can use the graph to represent the electrical network in the physics,and we will call that a network graph.In this thesis,we consider the resistance distances between two vertices of the ring type network graph,the path type network graph and the Sierpinski gasket network graph with dimension two.Let m₁,m₂,···,m_nbe n positive integers,V=V₁?V₂?···?V_n,E={uv|u?V_i,v?V_i+1,i=1,2,···,n},where i?=j,V_i?V_j=?|V_i|=m_i,V_n+1=V₁.We denote the ring type network graph G[m_i]₁ⁿand the path type network graph P[m_i]₁ⁿwith both vertex sets V and edge sets E and E\{uv|u?V₁,v?V_n},respectively.Using the principle of substitution,the triangle-star transformation and the series and parallel formula,we express arbitrary two vertices resistance distances of the ring type graph and path type graph in terms of the number of their vertices,respectively.Using the same formula,we derive some two-vertex resistance distances of the Sierpinski gasket network graph with dimension two.