## Vertex-wise invariants of vertex-weighted graphs: When can we make them all equal?

This question is very broad, but I like it that way. Let $G$ be a graph with positive vertex weights $\lbrace w(v) | v\in V(G) \rbrace$. We now want to consider a nonnegative vertex-wise function $f(v,w) = f_v(w)$.

$f_v(w)$ is to be defined in a graph theoretic way, e.g. 1. $f_v(w)$ is the total weight on the closed neighbourhood of $v$; e.g. 2. $f_v(w)$ is the probability that $v$ is in a maximum-weight stable set chosen according to such and such probability distribution.

My question is this: Under what conditions on $f$ can we guarantee the existence of a weight function $w$ such that $f(v,w)$ is equal for every vertex $v$?

An obvious example of a sufficient condition is: for every $v$, $f(v,w)$ is a continuous unbounded positive function of $v$, e.g. $f(v,w)=w$ for all $v$.

An obvious "unequalizable" function would be the first example, i.e. the total weight on the closed neighbourhood. If the closed neighbourhood of $v$ is a proper subset of the closed neighbourhood of $u$, then $f(u,w)>f(v,w)$ for any nonnegative weight function $w$.

So my question is, (1) does anybody know of any relevant research in this area? (2) Does anybody know of any interesting sufficient conditions? I am particularly interested in conditions that use the local structure of the graph.

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