What are current trends in Applied Graph Theory? I am interested mainly in non-algorithmical problems. Maybe even in applications of graphs to other mathematical disciplines. For example, abstract algebra naturally produces the notion of non-commuting graph of a group and prime graph of a ring.

Also, what about modern Chemical Graph Theory? Is it still bounded by finding upper and lower bounds on graph indexes like Wiener's, Zagreb's, Randic's and other?

Finally, what graph-theoretical problems arose say from biological networks or from the study of social graphs?

Clearly I am not specific on this question. Sorry for that! However, I have strong interest in graphs applications, so I had to ask mathoverflowers something like that! Also, thanks for your answers!