The process is replacing every value in a graph with the sum of the values of the vertices connected to it.
As a simple example, in a 3-cycle graph, it would look like this:
I am looking for values (for x, y and z) which scale by a constant when this transformation is applied to them. For small graphs like this I can easily find possible constants with simultaneous equations:
$$\lambda x = z + y\\\lambda y = z + x\\\lambda z = x + y\\$$
which gives $\lambda = -1$ if $x + y + z = 0$, and $\lambda = 2$ if $x = y = z$.
However, for much larger graphs, this takes much more time. Can these specific scalars that are different for most graphs be calculated from the structure of the graph directly, and what do they have to do with the graph? Also, do they have a name that I could use to look up things about them?
edit: is there also a name for the same thing but with multiplying the adjacent nodes instead of adding?