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For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

Here is a simple example of a chip-firing clock. Consider the graph with vertices $1,\dots, n$ and edges from vertex $i$ to $i+1$ and $n$ to $1$.

Then the function of a vector $a_1,\dots, a_n$ given by $f(a_1,\dots,a_n) = \sum_i i a_i \mod n$ is a chip-firing clock.

In this case, the cokernel of the Laplacian is $\mathbb Z$, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is $n$.

For a more complicated example, consider a graph with vertices $1,\dots n$ where vertex $i$ has edges to $i+1$ and $n$ except for $n$ which just connects to $n+1$.

Then the function of a vector $a_1,\dots a_n$ given by $f(a_1,\dots, a_n) = \sum_{i=1}^n 2^i a_i \mod (2^n-1)$ is a chip-firing clock. Again this is an example where the cokernel of the Laplacian is just $\mathbb Z$.

For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in image of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

Here is a simple example of a chip-firing clock. Consider the graph with vertices $1,\dots, n$ and edges from vertex $i$ to $i+1$ and $n$ to $1$.

Then the function of a vector $a_1,\dots, a_n$ given by $f(a_1,\dots,a_n) = \sum_i i a_i \mod n$ is a chip-firing clock.

In this case, the cokernel of the Laplacian is $\mathbb Z$, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is $n$.

For a more complicated example, consider a graph with vertices $1,\dots n$ where vertex $i$ has edges to $i+1$ and $n$ except for $n$ which just connects to $n+1$.

Then the function of a vector $a_1,\dots a_n$ given by $f(a_1,\dots, a_n) = \sum_{i=1}^n 2^i a_i \mod (2^n-1)$ is a chip-firing clock. Again this is an example where the cokernel of the Laplacian is just $\mathbb Z$.

For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

Here is a simple example of a chip-firing clock. Consider the graph with vertices $1,\dots, n$ and edges from vertex $i$ to $i+1$ and $n$ to $1$.

Then the function of a vector $a_1,\dots, a_n$ $f(n) = \sum_i i a_i \mod n$ is a chip-firing clock.

In this case, the cokernel of the Laplacian is $\mathbb Z$, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is $n$.

For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

Here is a simple example of a chip-firing clock. Consider the graph with vertices $1,\dots, n$ and edges from vertex $i$ to $i+1$ and $n$ to $1$.

Then the function of a vector $a_1,\dots, a_n$ given by $f(a_1,\dots,a_n) = \sum_i i a_i \mod n$ is a chip-firing clock.

In this case, the cokernel of the Laplacian is $\mathbb Z$, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is $n$.

For a more complicated example, consider a graph with vertices $1,\dots n$ where vertex $i$ has edges to $i+1$ and $n$ except for $n$ which just connects to $n+1$.

Then the function of a vector $a_1,\dots a_n$ given by $f(a_1,\dots, a_n) = \sum_{i=1}^n 2^i a_i \mod (2^n-1)$ is a chip-firing clock. Again this is an example where the cokernel of the Laplacian is just $\mathbb Z$.

For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function.

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction.

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.

Here is a simple example of a chip-firing clock. Consider the graph with vertices $1,\dots, n$ and edges from vertex $i$ to $i+1$ and $n$ to $1$.

Then the function of a vector $a_1,\dots, a_n$ $f(n) = \sum_i i a_i \mod n$ is a chip-firing clock.

In this case, the cokernel of the Laplacian is $\mathbb Z$, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is $n$.