If two vertices are non-adjacent in Petersen graph then they have exactly one comon neighbour.

Question

How do I prove that if two vertices are non-adjacent in Petersen graph then they have exactly one common neighbour.

Answer 1

This rather depends on your definition of the Petersen graph. One definition is that the ten vertices are labelled $P_A$ where $A$ runs through the two-element subsets of $\{1,2,3,4,5\}$, and that $P_A$ and $P_B$ are adjacent iff $A\cap B =\emptyset$. If $P_A$ and $P_B$ are non-adjacent, then $|A\cup B|=3$, and the only $P_C$ adjacent to both is with $C=\{1,2,3,4,5\}\setminus(A\cup B)$.