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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.04005 |
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| _version_ | 1866910805095612416 |
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| author | Kirkland, Stephen Li, Yuqiao McAlister, John Zhang, Xiaohong |
| author_facet | Kirkland, Stephen Li, Yuqiao McAlister, John Zhang, Xiaohong |
| contents | Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures the average travel time for a random walk to reach a randomly selected vertex. It is known that when an edge is added to $G$, the value of Kemeny's constant may either decrease, increase, or stay the same. In this paper, we present a quantitative analysis of this behaviour when the initial graph is a tree with $n$ vertices. We prove that when an edge is added into a tree on $n$ vertices, the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$ while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We also identify the trees, and the edges to be added, that correspond to the maximum increase and maximum decrease. Throughout, both matrix theoretic and graph theoretic techniques are employed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04005 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Edge Addition and the Change in Kemeny's Constant Kirkland, Stephen Li, Yuqiao McAlister, John Zhang, Xiaohong Combinatorics 05C50, 60J10 Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures the average travel time for a random walk to reach a randomly selected vertex. It is known that when an edge is added to $G$, the value of Kemeny's constant may either decrease, increase, or stay the same. In this paper, we present a quantitative analysis of this behaviour when the initial graph is a tree with $n$ vertices. We prove that when an edge is added into a tree on $n$ vertices, the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$ while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We also identify the trees, and the edges to be added, that correspond to the maximum increase and maximum decrease. Throughout, both matrix theoretic and graph theoretic techniques are employed. |
| title | Edge Addition and the Change in Kemeny's Constant |
| topic | Combinatorics 05C50, 60J10 |
| url | https://arxiv.org/abs/2306.04005 |