Given $\ell\ge 1$, we say a graph $G$ is $\ell$-good if for each $u,v\in G$ (not necessarily distinct), the number of walks of length $\ell$ from $u$ to $v$ is odd. We say a graph $G$ is good if it is $\ell$-good for some $\ell\ge 1$.

Do good graphs exist? For clarity, I am only talking about simple graphs (which lack loops and multiple edges).

**Context:** In Stanley’s book on Algebraic combinatorics, Exercise 1.13 is about proving an interesting property held for all good graphs. A friend of mine told me that after solving the exercise, he realized he didn’t know of any example of such graphs. I too am stumped about whether such graphs can exist.

A computer search revealed that none exist with $7$ or fewer vertices. I am unclear about the specifics of the search, they were done by my friend.

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