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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$s? Are there any definitions of compositions without the Lexicographic product? 2 added 93 characters in body; edited title # LexicographicproductsComposition of weighted multiedge digraphs Consider$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? 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?