Universal diameter bounds for random graphs with given degrees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918232146837504 |
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| author | Addario-Berry, Louigi Crudele, Gabriel |
| author_facet | Addario-Berry, Louigi Crudele, Gabriel |
| contents | Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so $\mathrm{diam}^+(G)=\infty$ if $G$ is not connected.
Fix a sequence $(d_1,\ldots,d_n)$ of positive integers, and let $\mathbf{G}$ be a uniformly random connected simple graph with $V(\mathbf{G})=[n]:=\{1,\ldots,n\}$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$. We show that, unless a $1-o(1)$ proportion of vertices have degree $2$, then $\mathbb{E}[\mathrm{diam}(\mathbf{G})]=O(\sqrt{n})$. It is not hard to see that this bound is best possible for general degree sequences (and in particular in the case of trees, in which $\sum_{v=1}^n d_v = 2(n-1)$). We also prove that this bound holds without the connectivity constraint. As a key input to the proofs, we show that graphs with minimum degree $3$ are with high probability connected and have logarithmic diameter: if $\min(d_1,\ldots,d_n) \ge 3$ and $\mathbf{G}$ is a uniformly random simple graph with $V(\mathbf{G})=[n]$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$, then $\mathrm{diam}^+(\mathbf{G})=$ $O_{\mathbb{P}}(\log n)$; this bound is also best possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal diameter bounds for random graphs with given degrees Addario-Berry, Louigi Crudele, Gabriel Probability Combinatorics 05C80, 60C05 Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so $\mathrm{diam}^+(G)=\infty$ if $G$ is not connected. Fix a sequence $(d_1,\ldots,d_n)$ of positive integers, and let $\mathbf{G}$ be a uniformly random connected simple graph with $V(\mathbf{G})=[n]:=\{1,\ldots,n\}$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$. We show that, unless a $1-o(1)$ proportion of vertices have degree $2$, then $\mathbb{E}[\mathrm{diam}(\mathbf{G})]=O(\sqrt{n})$. It is not hard to see that this bound is best possible for general degree sequences (and in particular in the case of trees, in which $\sum_{v=1}^n d_v = 2(n-1)$). We also prove that this bound holds without the connectivity constraint. As a key input to the proofs, we show that graphs with minimum degree $3$ are with high probability connected and have logarithmic diameter: if $\min(d_1,\ldots,d_n) \ge 3$ and $\mathbf{G}$ is a uniformly random simple graph with $V(\mathbf{G})=[n]$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$, then $\mathrm{diam}^+(\mathbf{G})=$ $O_{\mathbb{P}}(\log n)$; this bound is also best possible. |
| title | Universal diameter bounds for random graphs with given degrees |
| topic | Probability Combinatorics 05C80, 60C05 |
| url | https://arxiv.org/abs/2507.10759 |