Key Insights on Graph Theory and Geometric Representations

The same difference that exists between a dynamic law and its geometric representation, such as between the quaternary and its symbolic representation through the geometric figure of the square. In reality, a graph has no predefined drawing or shape a priori, but is a purely abstract reality formed exclusively by points and relations between points. When we want to resort to a formal representation to represent this geometric reality, we associate it with an image in which - canonically - individual entities are represented by points and the relations between them are represented by lines. This formal representation of the graph is called graph drawing. However, since originally a graph does not specify the type of relationship that connects the points to each other, the type of path necessary to pass from one point to another is totally arbitrary. The same link between two points can be represented as rectilinear, curved, broken, of arbitrary length and still remain a faithful formal representation of the graph. The same identical graph can therefore be represented by distinct drawings which, apparently, may have little in common, but in reality simply express different formal incarnations of the same abstract geometric entity. Each drawing is a particular expression of a general aspect contained in the original graph. Among all possible graph drawings, however, there are some that cover particular interest as they manage to manifest more explicitly the intrinsic qualities of the graph that others instead hide. In the two figures relating to the Petersen graph (Fig. 6) we can see how two distinct drawings of the same graph can highlight two different symmetries: the specular one in the first case and the pentagonal one in the second. Both of these symmetries are inherent characteristics of the Petersen graph, therefore present in each of its possible drawings. However, in some of these they are perfectly manifested and can therefore be easily grasped by the observer. Currently an entire research sector of graph theory studies the most efficient drawing methods to trace symmetric structures within graphs.

Line Graph. An important element for the study of a graph is constituted by its line graph. The latter is obtained by assigning a point to each line of the first and connecting two points to each other only in the case where the two lines shared at least one point in common in the starting graph. If the original graph was indicated with the letter $G$, this new graph constructed from the first will be indicated as $L(G)$. From a mathematical point of view, the line graph is a construction that allows us to better understand the relationships that exist between the individual nodes or elements of the graph. In this construction, in fact, the original subjects (the points) lose importance in relation to the relationships themselves. From a purely symbolic point of view, the line graph represents a way to extract and make explicit the meaning or behavior of a graph. In a certain way it represents the development of the graph, the explanation of its dynamics. The subject of investigation, in fact, is no longer the points, but their mutual interaction, which originally was represented by the lines of the graph. Symbolically, therefore, the construction of this graph coincides with a change in the investigation plane whose purpose is to see the dynamic developments inherent in a geometric configuration. This tool can therefore be helpful in the analysis of a graph, a bit like, from an arithmetic point of view, the analysis of the triangular of a number or the development of a sequence could be helpful.

Example of symbolic analysis of Line Graphs. As an example we can analyze the succession of line graphs of $K_{1,5}$ that is of a star formed by 5 external vertices connected to each other by a single central point. Symbolically the entire succession is synthesized by the initial star that defines its subject.