## Composition of weighted multiedge digraphs

Consider $G$ and $H$, two weighted multiedge digraphs with edge weights $\{g_{i}\}$${i=1}^{|E{G}|} and \{h_{j}\}$${j=1}^{|H{G}|}$

respectively where $|E_{G}|$ and $|E_{H}|$ are total number of edges in graphs $G$ and $H$ respectively and $g_{i},h_{j} \in \mathbb{R}$.

What is correct definition of a(n associative) composition of $G$ and $H$? Is there a definable notion of adjacency and biadjacency matrices in terms of the constituent multiedged weighted digraphs?

Will the composition be associative if the weights have $\pm \infty$ and $0$s?

Are there any definitions of compositions without the Lexicographic product?

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