Steffi’s graphs

Steffi, Steffi, the queen of Wimbeldon, steps on the grass court to play against a simpleton. The simpleton thinks that he is strong, but Steffi knows that his graphs are wrong. Crack! Crack! The spectators hear a sound and find the simpleton on the ground. Steffi approaches him across the net, his shorts are torn she can bet. The simpleton surrenders and wonders, how can Steffi’s shots be like thunders.

Steffi tells him, it’s because she hath, a hidden power called math. Compared to her, his powers are half, because he does not know Steffi’s graphs. On Earth’s crust, the upper strata, a graph in graph theory bears no relation to graphs that chart the data. Throw your graph paper in river Missouri, if you want to understand the graph theory.

"In graph theory," Steffi opines, "a graph is a collection of dots that may or may not be connected to each other by lines. By the moon and the star, it doesn’t matter how big the dots are. It does not matter how long or how straight is the line. If the dots are not round, it’s just fine." Graf opines, "All that matters is which dots are connected by which lines. If two dots are connected by one line regal, to draw another line connecting these is not legal."

In places urban and rural, each dot is called a vertex and vertices is its plural. Vertices are connected by lines called edges and edges can connect any or all of the vertices. If you are wise, the number of vertices a graph has, will tell you its size. (A graph is regular if all of its vertices have the same degree. The degree of a vertex is the number of edges that touch it.)

"Steffi, can your graphs eat great danes?"

"No, but these can plan routes of airplanes. If you are a master hack, these help you design a code that no one can crack. Whether we are sitting or taking a bath, we can plan a flight route along hamiltonian path."

(A hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A hamiltonain path, however, does not necessarily pass through all the edges of the graph. A hamiltonian path that ends in the same place where it began is called a hamiltonian circuit or a hamiltonain cycle.)

The connecting dots and lines of a graph are Steffi’s tutor, whenever she has to design a computer. (On the smallest end of the scale, a graph can be used to model the way tiny pulses of electricity flow through silicon chips that are inside electronic devices. In the big picture, a graph can model the ways computing systems can be interconnected, even when these are located all over the world.)

Steffi gives simpleton a task so simple, yet so hep — to lay out where the electric impulses should go on a silicon chip. Simpleton is allowed to use only one side of the chip, because the other side is for some other trick. If any of the connections cross, the whole thing will short-circuit and anger his boss. Steffi tells him not to fight, but to draw the graph that represents connections that are right.

This graph will be of diameter three, but it will also be planar, you see. (That means that it can be drawn in a flat plane so that edges only touch each other at a vertex. Specifications about the relationship between degree and diameter in graphs are used often while designing computer systems.)

Steffi says that thoughts are free, the figures here are graphs, too, but you won’t agree. She takes out a needle and a thread and gives it to simpleton. She says, "Simpleton, the uses of these graphs are lots, but this is just for your shorts."

— Aditya Rishi