In graph theory, Let $A$ be adjacent matrix, a vector $v$ can be represented by numbers on each vertices. $v_i$ denotes the number(or label) of node $i$.
Now I want this vector $u$, such that $u_i$ counts the numbers neighbors $j$ of node $i$ who satisfies $v_j \geq v_i$, i.e. $u_i=| \{\;j:A_{ij}=1 \wedge v_j \geq v_i \; \}|$.
Using $A$ and $v$ and some other graph matrix(Laplacian and so on), how to get $u$ in matrix equations, pure math, not matlab commands?
Suppose this matrix operations $f(A,v)$ produce $u$, I want the solution $v$ such that $f(A,v) = v$(I want $u$ be equal to $v$). What equation should this be look like? Can I cast this into linear systems to get a solution?
Thanks.