### Abstract

Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdos-Rényi random graphs. Since the graphs are random, their Kirchhoff indices are random variables. We give formulas for the expected value of the Kirchhoff index and show it concentrates around its expectation. We achieve this by studying the trace of the pseudoinverse of the Laplacian of Erdos-Rényi graphs. For synchronization (a class of estimation problems on graphs) our results imply that acquiring pairwise measurements uniformly at random is a good strategy, even if only a vanishing proportion of the measurements can be acquired.

Original language | English (US) |
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Pages (from-to) | 74-80 |

Number of pages | 7 |

Journal | Systems and Control Letters |

Volume | 74 |

DOIs | |

State | Published - Dec 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Computer Science(all)
- Mechanical Engineering
- Electrical and Electronic Engineering

### Keywords

- Cramér-Rao bounds
- Erdos-Rényi
- Estimation on graphs
- Kirchhoff index
- Laplacian Random matrices
- Pseudoinverse of graph
- Resistance distance
- Sensor network localization
- Synchronization

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## Cite this

*Systems and Control Letters*,

*74*, 74-80. https://doi.org/10.1016/j.sysconle.2014.10.006