Resistance matrices of balanced directed graphs. (English) Zbl 1485.05071
Summary: Let \(G\) be a strongly connected and balanced directed graph. We define the resistance \(r_{ij}\) between any two vertices \(i\) and \(j\) of \(G\) by using the Moore-Penrose inverse of the Laplacian matrix of \(G\) and define the resistance matrix by \(R:=[r_{ij}]\). This generalizes the resistance in the undirected case. In this paper, we show that \(R\) is a non-negative matrix and obtain an expression to compute the inverse, determinant and cofactor sums of \(R\).
MSC:
| 05C20 | Directed graphs (digraphs), tournaments |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
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