A Simple Method for Computing Resistance Distance

Abstract

The resistance distance r i j between two vertices v i and v j of a (connected, molecular) graph G is equal to the effective resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any edge is unity. We show how r i j can be computed from the Laplacian matrix L of the graph G: Let L(i) and L(i, j) be obtained from L by deleting its i-th row and column, and by deleting its i-th and j-th rows and columns, respectively. Then r i j = det L(i, j)/ det L(i).

The unicyclic graphs with maximum degree resistance distance

“…The resistance distance between two vertices u and v of graph G, denoted by R G (u, v), is originally defined to be the effective resistance between the corresponding two nodes u and v in the electrical network. More results and devolvement can be found in [1][2][3]25,36,42].…”

“…We need to consider two cases: (1) if v does not belong to the unique cycle C, then v will be a pendent vertex after using Transformation I, which yields that D v (G) ≤ n 2 − 19 3 ; (2) if v lies in the cycle C, by using Transformation II, one can obtain a new unicyclic graph containing a cycle C (|C | < |C|) such that the vertex v is not belonging to the cycle C . Thus the discussion will go back to case (1).…”

Visualizing the Electrical Structure of Power Systems

“…Finally, note that resistance distances are closely related to random walks on graphs [69], and to the eigenvalues and eigenvectors of the graph's Laplacian, and they can be calculated by such means [70][71][72]. This is an example of spectral graph theory [54], whose application to power system problems has only recently emerged [35,61,[73][74][75][76].…”

Resistance and relatedness on an evolutionary graph

“…where the G ab is the a,b entry of G. More conveniently, the resistance distance may be calculated with a theorem of Bapat et al [26]:…”