Let $G$ be an undirected (weighted) graph with non-negative edges.
When is it possible to label the vertices with strictly positive real numbers, such that the sum of labels of any vertex' neighbors is equal to some constant $c$?
Of course, if such a labelling exists, it is only unique up to positive scaling.
The closest concept I found so far was that of a distance magic labelling. Yet it requires a bijective mapping between the vertices and natural numbers. I do not require injectivity, and allow for arbitrary positive real numbers. Is there a name/literature on this kind of problem?
It obviously works with $k$-regular graphs, where I simply assign the same number to every vertex. From this I conjecture that the existence of such a labelling necessitates that $j$, the all-one vector, lies in the conic hull of the corresponding adjacency matrix' column space. This is violated for the graph $P_5$, for example, where no such labelling can exist (even allowing for negative real numbers, but not the zero label).
I am specifically interested in conditions on the adjacency matrix for such a labelling to exist. Furthermore, what changes when a neighbor's label is weighted by the weight of the shared edge?
This is true. We can always scale so that $c=1$, and a labeling $x$ (thought of as a $|V(G)|$-dimensional vector) makes the neighbors of every vertex sum to $1$ if and only if $Ax = j$, where $A$ is the adjacency matrix of the graph. (The same thing holds for the weighted adjacency matrix, if $G$ is a weighted graph.)