Exploring the geometric ratios in Rick’s paradox. When we first started looking at non-linear structures, we learned about their most fundamental characteristic: that their data doesn’t follow an order — at least, not an obvious numerical one, like we see in arrays or linked lists. As it turns out, this is the case with graphs. Nodes can be connected in any way possible, really. %PDF-1.4 People needed a good way of labeling and processing tasks, so they found a way to build upon the tools they had and created a way to juggle all the things that one single system needed to do, at any given time. With graphs, all of these restrictions go straight out the window. We could represent Twitter as a directed graph. Well, for one thing, a tree can only flow in one direction, from the root node to either leaf nodes or child nodes. This is called the complete graph on ve vertices, denoted K5; in a complete graph, each vertex is connected to each of the others. And trust me, some graphs really are complicated! In a directed edge, we can only travel from the origin to the destination, and never the other way around. So what happens when I follow you back? Instead, let’s look at the two types of graphs that are pretty easy to spot, and also pretty common in graph theory problems: directed graphs, and undirected graphs. And…why should we care? The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, … A tree with ‘n’ vertices has ‘n-1’ edges. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The formal, mathematical definition for a graph is just this: G = (V, E). Well, now that we’re experts in tree data structures, it only seems right to understand where trees came from. I’ve talked about abstractions a whole lot in this series, because ultimately, that’s what this series is about: finding the joy in the abstractions that lie beneath the things that all of us use, every single day. [�%O�lF@b%�j���xf�Q }�@t��K� �x �D8�FI However, it’s an entirely different story with undirected edges. And why would they? Well, it works because that ordered pair — (V, E) — is actually made up of two objects: a set of vertices, and a set of edges. The “unordered” part is really important here, because remember, unlike trees, there is no hierarchy of nodes! So, what is it that makes a tree different from the large umbrella of graphs? Time to change that. I certainly didn’t cover enough information here to fill a book, but that doesn’t mean you can’t keep learning about graphs! Go figure, right? On Facebook, if I add you as a friend, you must accept my request. In an undirected edge, the path that we can travel goes both ways. stream If this is your very first foray into discrete math, fear not — it’s mine, too! Modeling Linear Population Growth with Matrices. is bidirectional. But what if we did something kind of crazy and just…threw these rules out the window? There’s no concept of an “origin” and a “destination” node — instead, you’re my friend and I am yours. So we’ll do just that. Edges (sometimes referred to as links) can connect nodes in any way possible. Is there any direction or flow? But hang on a second — what if our graph has more than one node and more than one edge! I promise. I can follow you, but you might not follow me back. �xㄦ�?~j'FH���gȾr����������P�k�q����u��2�)�p!s�a",اVU�5SGYR�8��B�d�b�åo��\.^. Well, you probably used one today. But it would be a whole lot clearer if I had an example and actually wrote out the definition of a graph! In the example below, we have an undirected graph, with 8 vertices, and 11 edges. 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