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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.02535 |
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| _version_ | 1866911028601683968 |
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| author | Afshari, B. Afshari, M. |
| author_facet | Afshari, B. Afshari, M. |
| contents | Let $G$ be a graph on $n$ nodes with algebraic connectivity $λ_{2}$. The eccentricity of a node is defined as the length of a longest shortest path starting at that node. If $s_\ell$ denotes the number of nodes of eccentricity at most $\ell$, then for $\ell \ge 2$, $$λ_{2} \ge \frac{ 4 \, s_\ell }{ (\ell-2+\frac{4}{n}) \, n^2 }.$$ As a corollary, if $d$ denotes the diameter of $G$, then $$λ_{2} \ge \frac{ 4 }{ (d-2+\frac{4}{n}) \, n }.$$ It is also shown that $$λ_{2} \ge \frac{ s_\ell }{ 1+ \ell \left(e(G^{\ell})-m\right) },$$ where $m$ and $e(G^\ell)$ denote the number of edges in $G$ and in the $\ell$-th power of $ G $, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02535 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eccentricity and algebraic connectivity of graphs Afshari, B. Afshari, M. Combinatorics 05C50, 15A18 Let $G$ be a graph on $n$ nodes with algebraic connectivity $λ_{2}$. The eccentricity of a node is defined as the length of a longest shortest path starting at that node. If $s_\ell$ denotes the number of nodes of eccentricity at most $\ell$, then for $\ell \ge 2$, $$λ_{2} \ge \frac{ 4 \, s_\ell }{ (\ell-2+\frac{4}{n}) \, n^2 }.$$ As a corollary, if $d$ denotes the diameter of $G$, then $$λ_{2} \ge \frac{ 4 }{ (d-2+\frac{4}{n}) \, n }.$$ It is also shown that $$λ_{2} \ge \frac{ s_\ell }{ 1+ \ell \left(e(G^{\ell})-m\right) },$$ where $m$ and $e(G^\ell)$ denote the number of edges in $G$ and in the $\ell$-th power of $ G $, respectively. |
| title | Eccentricity and algebraic connectivity of graphs |
| topic | Combinatorics 05C50, 15A18 |
| url | https://arxiv.org/abs/2407.02535 |