MathDB

The Kingdom of Moles consists of

2023

cities, and there are several tunnels connecting different pairs of cities. Each tunnel allows bidirectional travel, and there is at most one tunnel connecting any two distinct cities. Furthermore, it is possible to travel between any two distinct cities through a combination of several tunnels.
A pair of distinct cities

\{A, B\}

is considered good if it satisfies the following condition for any city

C

different from

A

and

B

:
Let

d_{C,A}

be the minimum number of tunnels required to travel from

C

A

, and let

d_{C,B}

be the minimum number of tunnels required to travel from

C

B

. For any such pair

\{A, B\}

, it is guaranteed that regardless of the paths

X

and

Y

chosen to travel from

C

A

(using

d_{C,A}

tunnels) and from

C

B

(using

d_{C,B}

tunnels), respectively,

X

and

Y

do not share any common tunnel.
Find the second largest possible value for the number of good pairs of cities. Note that the pairs

\{A, B\}

and

\{B, A\}

are considered the same combination.

Problem Statement