Graph Theory and Symbolic Interpretations

In this case the subject is represented by the number 5 as an expression of 1. In this case, the five vertices represent 5 distinct active expressions of 1 which originated them, while 1 represents the secret source of their strength. We have called this source secret, insofar as one can notice that, although it is present in the first graph, in every subsequent graph it is hidden and invisible. This source represented by 1 is indeed transcendent and hierarchically superior to the other 5 vertices, therefore it is no longer visible in any of the successive manifestations, although it remains as the central point around which all figures are structured.

If, proceeding in our symbolic analysis, we consider 5 as a symbol of Universal Man and interpret this succession in a messianic key, then the first graph, that is $K_{1,5}$, represents the subject: the relationship between Man and divine immanence, represented by the central point. This first graph represents the subject of the subsequent discourse and is to be considered as outside of time.

The second graph $L(K_{1,5})$ will instead represent the first development or the first manifestation of the treated subject, in this case the state before the fall, the natural union between Man and Divinity. The third graph $L(L(K_{1,5}))$ will represent Man's distancing from God and his fall, while the fourth and last graph $L(L(L(K_{1,5})))$ will represent the final redemption and the restoration and perfection of the Human-Divine relationship.

This is an example of symbolic interpretation realized through line graphs. Clearly it is not the only one, other starting points will give rise to other considerations and reflections regarding these figures, all potentially contained in that unique geometric configuration of the initial star formed by 1 and 5 points.

Complement of a graph. Another common operation is that of constructing the complement graph. Given a graph $G$, its complement $\overline{G}$ is nothing other than the graph formed by the same vertices but with edges being the edges missing from the previous one. In practice, in $\overline{G}$ two points will be connected by a line if and only if in the original graph $G$ they are not. If we unite a graph to its complement we always obtain a complete graph, that is a graph whose number of edges is the maximum possible.

From a symbolic point of view, the complement of a graph produces a graph that is qualitatively related to the starting graph, but clearly complementary in its expression. If for example the starting graph expresses the action of number in its active aspect, the complement will represent it in its receptive aspect and vice versa.

Complete Graphs. A graph with the maximum number of possible edges is called a complete graph and is indicated as $K_n$, where the letter $n$ represents the number of vertices present in the graph. In particular, a complete graph of $n$ vertices will have a number of edges equal to $\binom{n}{2} = \frac{n(n-1)}{2}$, that is the triangular number of the preceding number. The simplest way to obtain a complete graph is to unite a graph to its complement graph. In doing so, the resulting graph will contain all possible edges and will therefore be complete.

From a mathematical point of view, complete graphs are extremely interesting elements as they constitute one of the founding pillars of modern geometry and algebra. Symbolically, complete graphs represent the plenitude of a number, its expressive fullness, the explication of all relationships contained within it. To better understand this aspect of complete graphs it is necessary to analyze their genesis.