This question is related to the question https://mathoverflow.net/questions/301626/what-is-known-about-graph-algebras but is not a duplicate, since here I am searching for examples of graph algebras.
It seems that "graph algebras" are not known in the literature, hence I am collecting examples of graph algebras.
I call a "graph algebra" a simple undirected graph $G=(V,E)$ and a binary mapping $+:E \rightarrow V$ such that:
(1) For all edges $(a,b)$ we have: $a+b \in N(a) \cap N(b)$, $N(a) = $ neighbors of $a$.
(2) $a+b=b+a$ for all edges
(3) $a+(b+c) = (a+b)+c$ for all edges $(a,b),(b,c)$ for which $(a,b+c),(a+b,c)$ are also edges.
[(4) if $a+b=a+c$ then $b=c$.]
I am collecting examples of such graph algebras. Examples of such "graph algebras" are:
a) infinite graph algebra: $V=\mathbb{N}$, $a \sim b : \leftrightarrow \gcd(a,b)=1$, $+:=+$ in $\mathbb{N}$
b) Let $H$ be a not necessarily finite or abelian group. Then $V=H$ and $a\sim b : \leftrightarrow a \cdot b = b \cdot a$, $+ := \cdot $ in $H$.
c) Let $G$ be a finite graph such that two adjacent vertices belong to an unique triangle and two non-adjacent vertices belong to an unique quadrilateral. Then if $(a,b)\in E$ there exists a unique $\phi(a,b)=\phi(b,a) \in V$ such that $(a,\phi(a,b)),(b,\phi(a,b))\in E$. Set $a+b := \phi(a,b)$. This makes the graph $G$ to a graph algebra.
d) Let $X$ be a topological space. $V:=\{ U \subset X | U \text{ is open set} \}$. Then $U \sim U' :\leftrightarrow U \cap U' \neq \emptyset$ and $U+U' := U \cup U'$. This defines a abelian graph algebra on the open sets of $X$ for which the cancellation law (4) does not necessarily hold.
e) Let $V:=H$ be a non-abelian group. Define $a\sim b : \leftrightarrow a\cdot b \neq b \cdot a$. And let $a * b := a \cdot b$ This defines a non abelian graph algebra for which the cancellation law (4) holds.
f) Let $V:=H$ be a real Hilbert space. Define $x\sim y : \leftrightarrow ( x,y) >0$, where $(x,y)$ is the dot-product. This defines a graph-algebra on the Hilbert space $H$, with $x \oplus y := x+y$.
g) Let $R$ be a commutative ring with $1$. Let $V:=$ zero divisors of $R$. Suppose that $R$ has the property (for example $\mathbb{Z}/(6 \mathbb{Z})$ has) that: $$ a,b,a-b \in V \rightarrow a+b \in V$$ Define $a\sim b :\leftrightarrow a-b \in V$ and $a+b:= a+b$ in $R$. This defines a abelian graph algebra on the zero divisors of $R$.
h) V=2-elements subsets of a set S, E : just one element in common, + : union minus the common element.
If somebody knows of other examples of graph algebras, that would be fine.
For example let $H:=D_4=$ dihedral group with $8$ elements. Then the corresponding graph in (b) is given by the picture below:
Another example: Let $R := \mathbb{Z} / ( 12 \mathbb{Z})$ and $V:= $ zero divisors of $R$. Then the graph of the graph algebra defined in g) is given by the picture below:
