I am attempting some practice questions where no answers are given. Would it be possible to verify my solution for this particular problem?

Let RBG be a coloured random graph with $n$ vertices $\{1, 2, · · · , n\}$, where every pair of vertices is connected either by a red edge (with probability $p$) or by a blue edge (with probability $q$) or by a green edge (with probability $r$), such that $p + q + r = 1$. (Note that all edges are present, but each edge carries exactly one of three different colours - red, blue or green)

A set of three vertices $\{i, j, k$} is said to form a “tri-coloured triangle” if the triangle formed by $\{i, j, k\}$ has exactly one red edge, exactly one blue edge and exactly one green edge. Let $N_3$ denote the number of tri-coloured triangles in the coloured random graph RBG. Calculate $E(N_3)$

My approach is as follows

Given a triangle, there are $3! = 6$ ways of colouring the triangle such that each edge has a distinct colour.

Therefore, $P(\text{A triangle is tri - coloured}) = 6pqr$.

There are ${n}\choose{3}$ ways of choosing 3 vertices in the random graph.

Therefore, $E(N_3) =$ ${n}\choose{3}$ $6pqr = n(n-1)(n-2)pqr$

  • $\begingroup$ Perfect, except for five missing full stops. $\endgroup$ – bof Dec 2 '18 at 4:55
  • $\begingroup$ Thank you, @bof ! :) $\endgroup$ – dzl Dec 3 '18 at 5:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.