Let us list down the shortest edge between each pair of vertices $x,y$ in the graph

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to y} \\\hline \text{1} & \text{2} & \text{2} \\\hline \text{1} & \text{3} & \text{there is a direct path from 1 to 3 of weight 8.We can also chose to go}\\ && \text{ via node 4 with total path weight $5+x.$ if $5+x<8(x<3)$ then shortest}\\&&\text{path is $5+x$ otherwise shortest path is the weight of the direct path}\\ && \text{i.e 8} \\\hline \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline \text{2} & \text{4} & \text{there is a direct path from 2 to 4 of weight 8. We can also chose to go}\\ && \text{via node 3 with total path weight}{ 5+x}\text{. If }{ 5+x<8(x<3)}\text{ the shortest}\\ && \text{path is 5+x. otherwise shortest path is the weight of the direct path}\\ && \text{i.e 8} \\\hline \text{3} & \text{4} & \text{We can chose to go through the direct path of weight x or via node }\\ && \text{to with weight } 5+8=13 \text{. If }{x<13}\text{ then we will chose x to be the shortest}\\&& \text{path otherwise $13$ is the shortest path} \ \\\hline \end{array}$$

- Case 1: Let us say $x < 3$. Say $x = 2$.

When we put $x = 2$ the above table is modified as

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to y} \\\hline \text{1} & \text{2} & \text{2} \\\hline \text{1} & \text{3} & \text{7} \\\hline \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline \text{2} & \text{4} & \text{7} \\\hline \text{3} & \text{4} & \text{2 // Note that the shortest path between nodes $3$ and $4$ is $x =2$} \\\hline \end{array}$$

- Case 2: $3 \leq x < 13$. Let's say $x = 12$. The table is now modified as

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to y} \\\hline \text{1} & \text{2} & \text{2} \\\hline \text{1} & \text{3} & \text{8} \\\hline \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline \text{2} & \text{4} & \text{8} \\\hline \text{3} & \text{4} & \text{12 // Note that the shortest path between nodes $3$ and $4$ is $x=12$} \\\hline \end{array}$$

Now the question asks you to find the largest possible integer value of $x$ such that shortest path between at least one pair of nodes in the graph is equal to $x.$ for values $x = 2,3,4,\ldots,12$ the shortest path between node $3$ and $4$ is equal to $x$.

The largest among this is $x = 12$. So the answer is $12$.