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On the Edge Derivative of the Normalized Laplacian with Applications to Kemeny's Constant
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abstract
In a connected graph, Kemeny's constant gives the expected time of a random walk from an arbitrary vertex $x$ to reach a randomly-chosen vertex $y$. Because of this, Kemeny's constant can be interpreted as a measure of how well a graph is connected. It is generally unknown how the addition or removal of edges affects Kemeny's constant. Inspired by the directional derivative of the normalized Laplacian, we derive the directional derivative of Kemeny's constant for several graph families. In addition, we find sharp bounds for the directional derivative of an eigenvalue of the normalized Laplacian and bounds for the directional derivative of Kemeny's constant.
Forward citations
Cited by 2 Pith papers
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The Derivative of Kemeny's Constant as a Centrality Measure in Undirected Graphs
The rate of change of a graph's Kemeny constant when an edge weakens gives an always-positive, cut-edge-safe edge centrality, and its unweighted version also scores non-edges for link prediction.
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Parking completions are $\mathbf{x}$-parking functions
For any fixed set of taken parking spots, the parking completions are precisely the x-parking functions whose cumulative bounds are the unoccupied spots.
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