The graph entropy was proposed by Körner in the year 1973 when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.