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In lack for a better name 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 $(a,b),(b,c)$

[(4) if $a+b=a+c$ then $b=c$.]

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 the squares = quadrilaterals with only one point in common can be added, and this gives a "graph algebra".

I wanted to know, if this structure which I call "graph algebra" is a known structure, and what is its name? What is known about such structure? (Reference request).

For example let $H:=D_4=$ dihedral group with $8$ elements. Then the corresponding graph in (b) is given by the picture below:

graph algebra

Edit: I found other examples of graph-algebras. In this question: https://math.stackexchange.com/questions/2805133/searching-for-examples-of-graph-algebras i am collecting other examples of graph algebras. So if you know of examples or can define them, it would be fine to answer in the other question.

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