Jelly,  18 16  15 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ2EÐḟœị³EƲ€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Jœcṗ2EÐḟœị³EƲ€S - Link: list of lists of integers, M; integer, n J - range of length of M = [1,2,3,...,length(M)] œc - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]] € - for each: Ʋ - last four links as a monad: ṗ2 - Cartesian power with 2 Ðḟ - discard if: E - all-equal (i.e. diagonal co-ordinates like [3,3]) œị - multi-dimensional index into: ³ - program's 3rd argument (1st input), M E - all equal? S - sum 

Jelly,  18 16  15 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ2EÐḟœị³EƲ€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Jœcṗ2EÐḟœị³EƲ€S - Link: list of lists of integers, M; integer, n J - range of length of M = [1,2,3,...,length(M)] œc - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]] € - for each: Ʋ - last four links as a monad: ṗ2 - Cartesian power with 2 Ðḟ - discard if: E - all-equal (i.e. diagonal co-ordinates like [3,3]) œị - multi-dimensional index into: ³ - program's 3rd argument (1st input), M E - all equal? S - sum 

Jelly,  18 16  15 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ2EÐḟœị³EƲ€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Updating...

Jœcṗ€2EÐḟ€œị⁸E€S - Link: list of lists of integers, M; integer, n J  - range of length of M = [1,2,3,...,length(M)] œc  - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]]   € - for each: ṗ 2 - Cartesian power with 2  € - for each:  Ðḟ - discard if:   E - all-equal (i.e. diagonal co-ordinates like [3,3])   œị - multi-dimensional index into (vectorises): ⁸  - chain's left argument, M  € - for each:   E - all equal?   S - sum 

Jelly,  18 16  15 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ2EÐḟœị³EƲ€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Jœcṗ2EÐḟœị³EƲ€S - Link: list of lists of integers, M; integer, n J - range of length of M = [1,2,3,...,length(M)] œc - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]] € - for each: Ʋ - last four links as a monad:  ṗ2 - Cartesian power with 2  Ðḟ -  discard if: E -  all-equal (i.e. diagonal co-ordinates like [3,3]) œị  -  multi-dimensional index into: ³ - program's 3rd argument (1st input), M E  -  all equal? S - sum 

Jelly,  18  16 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ€2EÐḟ€œị⁸E€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Jœcṗ€2EÐḟ€œị⁸E€S - Link: list of lists of integers, M; integer, n J - range of length of M = [1,2,3,...,length(M)] œc - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]] € - for each: ṗ 2 - Cartesian power with 2 € - for each: Ðḟ - discard if: E - all-equal (i.e. diagonal co-ordinates like [3,3]) œị - multi-dimensional index into (vectorises): ⁸ - chain's left argument, M € - for each: E - all equal? S - sum 

Jelly,  18 16  15 bytes

Assumes self-connections may be any weight (i.e. they are not necessarily only either \$0\$ or the equal weight).

Jœcṗ2EÐḟœị³EƲ€S 

Try it online!
the two subsets of size 3 being ACE and BCD
With n=2 all 10 subsets of size 2 work as its symmetric.

How?

Updating...

Jœcṗ€2EÐḟ€œị⁸E€S - Link: list of lists of integers, M; integer, n J - range of length of M = [1,2,3,...,length(M)] œc - Combinations of length n e.g. [[1,2,3],[1,2,4],[1,3,4],[2,3,4]] € - for each: ṗ 2 - Cartesian power with 2 € - for each: Ðḟ - discard if: E - all-equal (i.e. diagonal co-ordinates like [3,3]) œị - multi-dimensional index into (vectorises): ⁸ - chain's left argument, M € - for each: E - all equal? S - sum