Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the classical concept of derivative?
Full version of the question.
Similarly, in 1970, Lowell W. Beineke calls this graph the derived graph of $G$ in his paper entitled Characterizations of derived graphs.
In 2017, a PhD dissertation entitled Structures of Derived Graphs was awarded to Khawlah Hamad Alhulwah at Western Michigan University. A large part of this work is about line graphs, and it gives the following intuition: "a derived graph of G is a graph obtained from G by a graph operation of some type."
It however seems to me that Sabidussi and Beineke (and, probably, others) had in mind a deeper connection between line graphs and the classical concept of (function) derivative.
In another line of works, the derived graph of $G$ has the same vertices as $G$ but they are connected by an edge if the distance between them in $G$ is $2$. This terminology was introduced in the 2009 paper Bounds On The Second Stage Spectral Radius Of Graphs by Ayyaswamy, Balachandran and Kannan, and further studied for instance in the 2012 paper Derived graphs of some graphs by Jog, Hande, Gutman, and Bozkurt. However, I can't find any explanation for this terminology in these papers.
My questions are:
- is there any reason to call line graphs graph derivatives? are they related to some extension of classical derivatives in a way that I miss?
- are there other definitions of graph derivatives that may be seen as such extensions?