Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R\_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C\_A(z) = Tr[R\_A(z) ]$. 


* I want to compare between the numbers $[R\_A(z)]\_{ii}$ and $\frac{1}{n} C\_A(z)$. 


Do we know as to when is which of them larger and by how much? 


Like given a $A$ and a $i$ I want to understand when it will be true that, $[R\_A(z)]\_{ii} \leq \frac{1}{n} C\_A(z)$ ?




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If necessary assume that $z > \lambda\_{max}(A)$


If necessary assume that $A$ is constructed as follows : First take the matrix $D - Ad$ where $D$ is the diagonal matrix of degrees of some bi-partite graph and $Ad$ is its adjacency matrix. Then flip some of the off-diagonal $-1$ entries of $D-A$ to $1$ keeping the entire thing symmetric. 



Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
The trace is the sum of the diagonal elements, so (when $z$ is real) it's always true for at least one $i$, and the only way it can be true for all of them is that all diagonal elements are equal. For example, this is the case for the matrix 
$$ A = \pmatrix{3 & 0 & 1 & -1\cr
 0 & 3 & 1 & -1\cr
 1 & 1 & 3 & 0\cr
 -1 & -1 & 0 & 3\cr} $$


