Resistance distance

In graph theory, the resistance distance between two vertices of a simple, connected graph, G, is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a resistance of one ohm. It is a metric on graphs.

Definition

On a graph G, the resistance distance Ω_i,j between two vertices v_i and v_j is^[1]

${\displaystyle \Omega _{i,j}:=\Gamma _{i,i}+\Gamma _{j,j}-\Gamma _{i,j}-\Gamma _{j,i},}$

where ${\displaystyle \Gamma =\left(L+{\frac {1}{|V|}}\Phi \right)^{+},}$

with ⁺ denotes the Moore–Penrose inverse, L the Laplacian matrix of G, |V| is the number of vertices in G, and Φ is the |V| × |V| matrix containing all 1s.

Properties of resistance distance

If i = j then Ω_i,j = 0. For an undirected graph

${\displaystyle \Omega _{i,j}=\Omega _{j,i}=\Gamma _{i,i}+\Gamma _{j,j}-2\Gamma _{i,j}}$

General sum rule

For any N-vertex simple connected graph G = (V, E) and arbitrary N×N matrix M:

${\displaystyle \sum _{i,j\in V}(LML)_{i,j}\Omega _{i,j}=-2\operatorname {tr} (ML)}$

From this generalized sum rule a number of relationships can be derived depending on the choice of M. Two of note are;

${\displaystyle {\begin{aligned}\sum _{(i,j)\in E}\Omega _{i,j}&=N-1\\\sum _{i<j\in V}\Omega _{i,j}&=N\sum _{k=1}^{N-1}\lambda _{k}^{-1}\end{aligned}}}$

where the λ_k are the non-zero eigenvalues of the Laplacian matrix. This unordered sum

${\displaystyle \sum _{i<j}\Omega _{i,j}}$

is called the Kirchhoff index of the graph.

Relationship to the number of spanning trees of a graph

For a simple connected graph G = (V, E), the resistance distance between two vertices may be expressed as a function of the set of spanning trees, T, of G as follows:

${\displaystyle \Omega _{i,j}={\begin{cases}{\frac {\left|\{t:t\in T,\,e_{i,j}\in t\}\right\vert }{\left|T\right\vert }},&(i,j)\in E\\{\frac {\left|T'-T\right\vert }{\left|T\right\vert }},&(i,j)\not \in E\end{cases}}}$

where T' is the set of spanning trees for the graph G' = (V, E + e_i,j). In other words, for an edge ${\displaystyle (i,j)\in E}$, the resistance distance between a pair of nodes ${\displaystyle i}$ and ${\displaystyle j}$ is the probability that the edge ${\displaystyle (i,j)}$ is in a random spanning tree of ${\displaystyle G}$.

Relationship to random walks

The resistance distance between vertices ${\displaystyle u}$ and ${\displaystyle u}$ is proportional to the commute time ${\displaystyle C_{u,v}}$ of a random walk between ${\displaystyle u}$ and ${\displaystyle v}$. The commute time is the expected number of steps in a random walk that starts at ${\displaystyle u}$, visits ${\displaystyle v}$, and returns to ${\displaystyle u}$. For a graph with ${\displaystyle m}$ edges, the resistance distance and commute time are related as ${\displaystyle C_{u,v}=2m\Omega _{u,v}}$.^[2]

As a squared Euclidean distance

Since the Laplacian L is symmetric and positive semi-definite, so is

${\displaystyle \left(L+{\frac {1}{|V|}}\Phi \right),}$

thus its pseudo-inverse Γ is also symmetric and positive semi-definite. Thus, there is a K such that ${\displaystyle \Gamma =KK^{\textsf {T}}}$ and we can write:

${\displaystyle \Omega _{i,j}=\Gamma _{i,i}+\Gamma _{j,j}-\Gamma _{i,j}-\Gamma _{j,i}=K_{i}K_{i}^{\textsf {T}}+K_{j}K_{j}^{\textsf {T}}-K_{i}K_{j}^{\textsf {T}}-K_{j}K_{i}^{\textsf {T}}=\left(K_{i}-K_{j}\right)^{2}}$

showing that the square root of the resistance distance corresponds to the Euclidean distance in the space spanned by K.

Connection with Fibonacci numbers

A fan graph is a graph on n + 1 vertices where there is an edge between vertex i and n + 1 for all i = 1, 2, 3, …, n, and there is an edge between vertex i and i + 1 for all i = 1, 2, 3, …, n – 1.

The resistance distance between vertex n + 1 and vertex i ∈ {1, 2, 3, …, n} is

${\displaystyle {\frac {F_{2(n-i)+1}F_{2i-1}}{F_{2n}}}}$

where F_j is the j-th Fibonacci number, for j ≥ 0.^[3]

See also

• Conductance (graph)

References

• Klein, D. J.; Randic, M. J. (1993). "Resistance Distance". J. Math. Chem. 12: 81–95. doi:10.1007/BF01164627. S2CID 16382100.
• Gutman, Ivan; Mohar, Bojan (1996). "The quasi-Wiener and the Kirchhoff indices coincide". J. Chem. Inf. Comput. Sci. 36 (5): 982–985. doi:10.1021/ci960007t.
• Palacios, Jose Luis (2001). "Closed-form formulas for the Kirchhoff index". Int. J. Quantum Chem. 81 (2): 135–140. doi:10.1002/1097-461X(2001)81:2<135::AID-QUA4>3.0.CO;2-G.
• Babic, D.; Klein, D. J.; Lukovits, I.; Nikolic, S.; Trinajstic, N. (2002). "Resistance-distance matrix: a computational algorithm and its application". Int. J. Quantum Chem. 90 (1): 166–167. doi:10.1002/qua.10057.
• Klein, D. J. (2002). "Resistance Distance Sum Rules" (PDF). Croatica Chem. Acta. 75 (2): 633–649. Archived from the original (PDF) on 2012-03-26.
• Bapat, Ravindra B.; Gutman, Ivan; Xiao, Wenjun (2003). "A simple method for computing resistance distance". Z. Naturforsch. 58a (9–10): 494–498. Bibcode:2003ZNatA..58..494B. doi:10.1515/zna-2003-9-1003.
• Placios, Jose Luis (2004). "Foster's formulas via probability and the Kirchhoff index". Method. Comput. Appl. Probab. 6 (4): 381–387. doi:10.1023/B:MCAP.0000045086.76839.54. S2CID 120309331.
• Bendito, Enrique; Carmona, Angeles; Encinas, Andres M.; Gesto, Jose M. (2008). "A formula for the Kirchhoff index". Int. J. Quantum Chem. 108 (6): 1200–1206. Bibcode:2008IJQC..108.1200B. doi:10.1002/qua.21588.
• Zhou, Bo; Trinajstic, Nenad (2009). "The Kirchhoff index and the matching number". Int. J. Quantum Chem. 109 (13): 2978–2981. Bibcode:2009IJQC..109.2978Z. doi:10.1002/qua.21915.
• Zhou, Bo; Trinajstic, Nenad (2009). "On resistance-distance and the Kirchhoff index". J. Math. Chem. 46: 283–289. doi:10.1007/s10910-008-9459-3. hdl:10338.dmlcz/140814. S2CID 119389248.
• Zhou, Bo (2011). "On sum of powers of Laplacian eigenvalues and Laplacian Estrada Index of graphs". Match Commun. Math. Comput. Chem. 62: 611–619. arXiv:1102.1144.

• Zhang, Heping; Yang, Yujun (2007). "Resistance distance and Kirchhoff index in circulant graphs". Int. J. Quantum Chem. 107 (2): 330–339. Bibcode:2007IJQC..107..330Z. doi:10.1002/qua.21068.

• Yang, Yujun; Zhang, Heping (2008). "Some rules on resistance distance with applications". J. Phys. A: Math. Theor. 41 (44): 445203. Bibcode:2008JPhA...41R5203Y. doi:10.1088/1751-8113/41/44/445203. S2CID 122226781.