**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.**

In a 1961 paper entitled *Graph Derivatives*, Gert Sabidussi defines the *graph derivative $\delta G$* of a graph $G$; it is nothing but the nowadays classical line graph of $G$.

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 derivativesthat may be seen as such extensions?

derivatives(not justderived) and denote them by $\delta G$ makes me unsure. $\endgroup$1more comment