***Short version:** in several papers, [line graphs](https://en.wikipedia.org/wiki/Line_graph) (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?*




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***Full version of the question.***


In a 1961 paper entitled *[Graph Derivatives](https://eudml.org/doc/169965)*, Gert Sabidussi defines the *graph derivative $\delta G$* of a graph $G$; it is nothing but the nowadays classical [line graph](https://en.wikipedia.org/wiki/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](https://doi.org/10.1016/S0021-9800(70)80019-9)*.


In 2017, a PhD dissertation entitled [*Structures of Derived Graphs*](https://scholarworks.wmich.edu/dissertations/3204/) 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](https://zenodo.org/record/1071578)* by Ayyaswamy, Balachandran and Kannan, and further studied for instance in the 2012 paper *[Derived graphs of some graphs](https://www.researchgate.net/publication/268015448_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?
> 
> 
> 



Line graphs called "graph derivatives": any intuition?
I just found the answer to this question in the paper *[Synthesis and analysis in total variation regularization](https://arxiv.org/abs/1901.0641)* by Francesco Ortelli and Sara van de Geer.


In the abstract, they write "*We give a definition of the discrete graph derivative operator based on the notion of line graph*", and they do so in Section 1.2. They use (a directed version of) the [incidence matrix](https://en.wikipedia.org/wiki/Incidence_matrix#Graph_theory) of a (directed) graph, and notice it may be used to differentiate a signal on the vertices of the graph along its edges, like [gmvh](https://mathoverflow.net/users/45250/gmvh) suggests in [the answer above](https://mathoverflow.net/q/389970). Then, in order to iterate such derivations, they introduce the (directed) line graph and say:



> 
> The idea behind using the line graph is the following: when computing
> differences along the direction of the graph, we obtain a value for
> each edge of the graph. Since we follow the direction of the edges,
> these values represent the first discrete derivative of the signal
> with respect to the graph, and constitute the vertices of the line
> graph.
> 
> 
> 


They finally define the *k-th order discrete graph derivative operator* from the incidence matrix of iterated line graphs (meaning that they take the line graph of the line graph of the line graph etc).


**It turns out that yes, line graphs and derivatives are strongly related!**


