# Random walks on graphs

@inproceedings{Sodin2004RandomWO, title={Random walks on graphs}, author={Sodin}, year={2004} }

Most of the material in this handout is taken from: Daniel Spielman, Spectral and Algebraic Graph Theory. Manuscript, 2019. Let A be the adjacency matrix of a graph G. We use α1 ≥ · · · ≥ αn to denote the eigenvalues of A (note that they are ordered in the opposite direction with respect to the eigenvalues λ1 ≤ · · · ≤ λn of the Laplacian matrix L). If G is d-regular, then L = I − 1 dA and therefore λi = 1− αi d . Since λi ∈ [0, 2] for any G (even not regular), we have that αi ∈ [−d, d] for any…

## 213 Citations

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### Spectra of Laplacian Matrices of Weighted Graphs: Structural Genericity Properties

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### APPROXIMATING GRAPH CONDUCTANCE: FROM GLOBAL TO LOCAL

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### Discrete Quantum Walks on the Symmetric Group

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- 2022

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### Algorithmic aspects of graph-indexed random walks

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It is shown that while the first problem is $\#$P-complete, the other two problems can be solved in polynomial time.

### Random walks on graphs with interval weights and precise marginals

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## References

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### Random Walks on Graphs: A Survey

- Mathematics
- 2001

Estimates on the important parameters of access time, commute time, cover time and mixing time are discussed and recent algorithmic applications of random walks are sketched, in particular to the problem of sampling.

### Random walks and electric networks

- Mathematics
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The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.

### Motwani and Raghavan

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### Grinstead and Snell

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