Finding a graph-theoretic representation of expressions in Boole's algebra

I just read "Boole's Algebra Isn't Boolean Algebra" by Theodore Halperin (behind a paywall here). I don't have a strong background in abstract algebra, so, frankly, the paper is a bit over my head but the gist of it is as follows: the algebra developed by Boole in the 19th century has some strange properties. Boole interprets every term in an expression as representing a set and restricts the domain of valid expressions on the basis of relationships between the underlying sets. In particular, he asserts that the expression $x + y$ is valid iff $x$ and $y$ represent disjoint sets, and $x - y$ is valid iff the intersection of $x$ and $y$ is nonempty. He also defines an operation $w = \frac x y$ such that $w$ has many solutions. Halperin goes to great lengths to construct a commutative ring that satisfies Boole's constraints.

I think that there might be a nice, intuitive, graph-theoretic interpretation of Boole's algebra where we can say something like "Given a complex expression in Boole's algebra, $\Phi = \{\phi_1, \phi_2\, \dots, \phi_n\}$, we can construct a (di)graph $G$ such that $\Phi$ is valid if and only if $G$ has some property $P$."

For example, to test if $X + Y$ is valid, we could do something like the following:

Let $G$ be a graph. Let $V(G) = X \cup Y \cup \{v_X, v_Y\}$, i.e. make a vertex for every element of the underlying set, plus a vertex for each term in the expression. Then define

$$E(G) = \{uv_X \mid u \in X \} \cup \{uv_Y \mid u \in Y \}$$

and let $\partial(G)$ be the set of all valid bonds formed by subsets of $V(G)$. It follows that $X + Y$ is valid if and only if $|\partial(G)| = 2$.

From there, I'm not sure how to go about forming graphs for complex expressions like $\frac {y(x + z)} {z^2 + 1}$ by composing simpler graphs. I imagine that there is literature about representing Boolean algebra on graphs, but I haven't been able to find it.

Does anyone have an elegant interpretation? Or a pointer to relevant literature that might ge

A bit of shameless self-promotion, but you might find http://www.sciencedirect.com/science/article/pii/0895717794001669 relevant, as well as the earlier paper I cite there by Frank Harary.

Those papers present the idea that a boolean function can be represented as a hypercube in which the vertices represent combinations of truth values. That is, rather than construct a graph with a vertex for each boolean variable X, Y etc, construct a graph where one vertex represents the tuple `, one represents ` etc.

StackExchange Computer Science Q#24088, answer score: 2