Assume $a_{ij}:=1+ c_{ij}$ is element of $n\times n$ symmetric matrices $A$ with diagonal element equals to $0$.
What condition we can impose on matrix $C$ (whose elements are $c_{ij}$), such that
$$pn^2+qn+r\leq \sqrt{2}\sum_{1\leq i<j\leq n}a_{ij}+\sum_{1\leq j\leq n}a_{ij}$$