Let $\kappa$ be an infinite cardinal and suppose $$n, d: \kappa \to \big((\kappa+1)\setminus \{0\}\big) = \{1, \ldots, \kappa\}$$ are arbitrary functions.


Is there $E \subseteq \big\{\{x,y\}: x\neq y \in \kappa\big\}$ such that the graph $G=(\kappa,E)$ has the following property?



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> > For all $k\in \kappa$ there are exactly $n(k)$ elements of $\kappa$ that have degree $d(k)$.
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Infinite graph with degrees given
You might find the following paper useful: [Degree sequences of infinite graphs, by Andreas Blass and Frank Harary](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135582/jlms0010.pdf).



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> The degree sequences of finite graphs, finite connected graphs, finite trees and finite forests have all
>  been characterized. Our present purpose is to provide such characterizations in the infinite case.
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