Graph theory is used to study the relations between objects. A graph is composed of vertices and edges in a diagram such as this:

A-----B

|    /   

|   /   

|  /     E

| /     /

|/     /

C-----D

In the above diagram, A is linked to B and C; B is linked to A, C, and E; C is linked to A, B, and D; D is linked to C and E; and E is linked to B and D. As that description was rather wordy, a graph can be represented as a symmetric boolean matrix where a 1 represents a connection and a 0 represents the lack thereof. The above matrix is translated to this:

01100

10101

11010

00101

01010

For the purpose of this problem, the matrix definition can be extended to include the distances or weights of the paths between nodes. If individual ASCII characters in the diagram have weight 1, he matrix would be:

05500

50502

55050

00502

02020

A "complete graph" consists of a set of points such that each point is linked to every other point. The above graph is incomplete because it lacks connections from A to D and E, B to D, and C to E. However, the subgraph between A, B, and C is complete (and equally weighted). A 4-complete graph would look like this:

A---B

| /|

| X |

|/ |

C---D

and would be represented by the matrix:

01111

10111

11011

11101

11110

This problem is as follows: Given a symmetric matrix representing a graph and a positive integer n, find the number of distinct equally-weighted complete subgraphs of size n contained within.

You may assume that the input matrix is numeric and symmetric, and may choose input/output format. An entry in the matrix may be part of multiple equally-weighted subgraphs as long as they are distinct and of equal size. You may assume that n is a positive integer greater than or equal to 3.

The winning criterion for this challenge is code golf. Standard rules apply.

Jelly, 16 bytes

ịⱮịŒDḊẎE

Jœcç€⁸S

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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

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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

Clean, 152 bytes

import StdEnv,Data.List

$m n=sum[1\i<-subsequences[0..size m-1]|length i==n,j<-permutations i|case[m.[x,y]\x<-i&y<-j]of[u:v]=all((==)u)v&&u>0;_=1<0]/2

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TIO driver takes n as a command-line argument and the matrix through STDIN (weights up to 9).

The actual function $ :: {#{#Int}} Int -> Int works with any size weights.