If in a graph, any two vertices have an odd number of common neighbors, is it true that, there exists a vertex which is a 'universal friend', i.e adjacent to every other vertex?
(It is known that in that case the number of vertices has to be odd:
If each of the odd numbers in question were 1, then this is the statement of the Friendship theorem.
As a counterexample, consider the Kneser graph $K(6,2)$: its vertex set consists of all $15$ subsets of $\{1,2,3,4,5,6\}$ of size $2$, and two vertices are connected whenever they correspond to disjoint subsets.
Then any two non-adjacent vertices have $3$ common neighbors: for example, $\{1,2\}$ and $\{1,3\}$ have common neighbors $\{4,5\}$, $\{4,6\}$, and $\{5,6\}$. Any two adjacent vertices have $1$ common neighbor: for example, $\{1,2\}$ and $\{3,4\}$ have common neighbor $\{5,6\}$.
However, there's no vertex adjacent to all other vertices; in fact, $K(6,2)$ is regular of degree $6$.
(Kneser graphs are often good counterexamples for things; the Petersen graph, which is $K(5,2)$, is the most popular of these. However, in this instance, I admit that I just searched all of the graphs in Mathematica's database until I found one with this property.)