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 1 ohm resistance. 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

\Omega_{i,j}:=\Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i}\,

where Γ is the Moore–Penrose inverse of the Laplacian matrix of G.

Properties of resistance distance

If i = j then

\Omega_{i,j}=0.\,

For an undirected graph

\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:

\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;

\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}

where the $\lambda_{k}$ are the non-zero eigenvalues of the Laplacian matrix. This unordered sum $Σ i<j Ω i,j$ 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 by expressed as a function of the set of spanning trees, T, of G as follows:

\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 $G'=(V, E+e_{i,j})$ .

As a squared Euclidean distance

Since the Laplacian $L$ is symmetric and positive semi-definite, its pseudoinverse $\Gamma$ is also symmetric and positive semi-definite. Thus, there is a $K$ such that $\Gamma = K K^T$ and we can write:

\Omega_{i,j} = \Gamma_{i,i}+\Gamma_{j,j}-\Gamma_{i,j}-\Gamma_{j,i} = K_iK_i^T + K_jK_j^T - K_iK_j^T - K_jK_i^T = (K_i - K_j)^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 $i \in \{1,2,3,...,n\}$ is $\frac{ F_{2(n-i)+1} F_{2i-1} }{ F_{2n} }$ where $F_{j}$ is the $j$ -th Fibonacci number, for $j \geq 0$ .^[1]^[2]

References

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