We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

Note: $M$ can be viewed as the adjacency matrix of a given undirected multi-edge graph. Alternatively, for all $1\le i<j\le n$, $M_{i,j}$ can be viewed as an integer capacity in a symmetric flow network. For all $1\le i<j\le n$, $z_{i,j}$ can be viewed as a function of the amount of "flow" between the two nodes $i$ and $j$ passing through paths of length at most $2$ (i.e., containing $2$ edges).

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

Note: $M$ can be viewed as the adjacency matrix of a given undirected multi-edge graph. Alternatively, for all $1\le i<j\le n$, $M_{i,j}$ can be viewed as an integer capacity in a symmetric flow network. For all $1\le i<j\le n$, $z_{i,j}$ can be viewed as a function of the amount of "flow" between the two nodes $i$ and $j$ passing through paths of length at most $2$ (i.e., containing $2$ edges).

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

Note: $M$ can be viewed as the adjacency matrix of a given undirected multi-edge graph. For all $1\le i<j\le n$, $z_{i,j}$ can be viewed as a function of the amount of "flow" between the two nodes $i$ and $j$ passing through paths of length at most $2$ (i.e., containing $2$ edges). Alternatively, for all $1\le i<j\le n$, $M_{i,j}$ can be viewed as an integer capacity in a symmetric flow network.

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

Note: $M$ can be viewed as the adjacency matrix of a given undirected multi-edge graph. Alternatively, for all $1\le i<j\le n$, $M_{i,j}$ can be viewed as an integer capacity in a symmetric flow network. For all $1\le i<j\le n$, $z_{i,j}$ can be viewed as a function of the amount of "flow" between the two nodes $i$ and $j$ passing through paths of length at most $2$ (i.e., containing $2$ edges).

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.

Let $z_{i,j}:=\frac{M_{i,j}}{\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\sum_{i<j}z_{i,j}$.

Question: Can we prove that the maximum value $z^*$ of $z$ over all matrices $M$ defined above is upper bounded by $\alpha n$ for some absolute constant $\alpha$?

Note: $M$ can be viewed as the adjacency matrix of a given undirected multi-edge graph. For all $1\le i<j\le n$, $z_{i,j}$ can be viewed as a function of the amount of "flow" between the two nodes $i$ and $j$ passing through paths of length at most $2$ (i.e., containing $2$ edges). Alternatively, for all $1\le i<j\le n$, $M_{i,j}$ can be viewed as an integer capacity in a symmetric flow network.