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Consider $G$ and $H$, two weighted multiedge digraphs with edge weights $\{g_{i}\}\{g_{i}\}$${i=1}^{|E{G}|}$ and $\{h_{i}\}\{h_{j}\}$${i=1}^{|Hj=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.0$s?

Are there any definitions of compositions without the Lexicographic product?
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Lexicographic products Composition of weighted multiedge digraphs

Consider $G$ and $H$, two weighted multiedge digraphs with edge weights

$\{g_{i}\}{i=1}^{|E{G}|}$ and

$\{h_{i}\}{i=1}^{|H{G}|}$

respectively where $|E_{G}|$ and $|E_{H}|$ are total number of edges in graphs $G$ and $H$ respectively.

What is correct definition of a(n associative) composition (Lexicographic product) of weighted multiedge digraphs? $G$ and $H$? Is there a definable notion of adjacency and biadjacency matrices in terms of the constituent multiedged weighted digraphs?Let the weights be variables $\{x_{i}\}_{i=1}^{|E|}$ where $|E|$ is the total number of multiedges in the digraph.

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

Are there any other definitions of compositions without the Lexicographic product?
show/hide this revision's text 1

Lexicographic products of weighted multiedge digraphs

What is correct definition of a(n associative) composition (Lexicographic product) of weighted multiedge digraphs? Is there a definable notion of adjacency and biadjacency matrices in terms of the constituent multiedged weighted digraphs? Let the weights be variables $\{x_{i}\}_{i=1}^{|E|}$ where $|E|$ is the total number of multiedges in the digraph.

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

Are there any other definitions of compositions without the Lexicographic product?