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

x |
y |
shortest path from x to y |

1 |
2 |
2 |

1 |
3 |
there is a direct path from 1 to 3 of weight 8.We can also chose to go via node 4 with total path weight 5+x. If 5+ x < 8 (x < 3) then shortest path is 5+x otherwise shortest path is the weight of the direct path i.e 8 |

1 |
4 |
5 |

2 |
3 |
5 |

2 |
4 |
there is a direct path from 2 to 4 of weight 8.We can also chose to go via node 3 with total path weight 5+x. If 5+ x < 8 (x < 3) then shortest path is 5+x otherwise shortest path is the weight of the direct path i.e 8 |

3 |
4 |
We can chose to go through the direct path of weight x or via node to with weight 5+8 = 13. If x < 13 then we will choose x to be the shortest path otherwise 13 is the shortest path |

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

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

x |
y |
shortest path from x to y |

1 |
2 |
2 |

1 |
3 |
7 |

1 |
4 |
5 |

2 |
3 |
5 |

2 |
4 |
7 |

3 |
4 |
2 // Note that the shortest path between node 3 and 4 is x = 2 |

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

x |
y |
shortest path from x to y |

1 |
2 |
2 |

1 |
3 |
8 |

1 |
4 |
5 |

2 |
3 |
5 |

2 |
4 |
8 |

3 |
4 |
12 // Note that the shortest path between node 3 and 4 is x =12 |

Now the question asks you to find the largest possible integer value of x such that shortest path between atleast one pair of nodes in the graph is equal to x. for values x = 2,3,4,....,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