The study is graph theory, and I won't link Wikipedia because its math pages are always overwhelming, other blog posts are likely more readable.

'Graph' in this sense does not mean these: https://royalsocietypublishing.org/cms/asset/ac7c91c1-a95b-4...

but like this: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n...

or this: https://lh3.googleusercontent.com/proxy/l31aliLTy0BqoatXyi7K...

It's what the geek-famous tool graphviz is for, visualising these kinds of graphs: https://duckduckgo.com/?t=ffab&q=dot+graphviz&iax=images&ia=...

The study is the connections between things, not the shape they make on paper. So not squares, triangles, hexagons, etc. but can you get from one thing to another and how many intermediate ones do you have to go through? Are there multiple paths from here to there, or just one? Which graphs have the same connectivity even when drawn in a different layout? What does it 'cost' to go from one to another (see below)?

It's used in the classic Konigsberg Bridge problem: https://physics.weber.edu/carroll/honors_images/BarbasiBridg... where the nodes are places in Konigsberg, and the bridges are the connections between them, and the puzzle is asking if you can visit all the areas, cross all the bridges once and only once, and return to where you started.

In the classic Travelling Salesman problem: https://cdn.optimoroute.com/wp-content/uploads/2020/07/Trave... where the salesman wants to visit all the cities, they certainly can use the same route more than once, but what's the most efficient route to visit them all without wasting time and fuel going back to the same one unnecessarily?

Edges can have weights (numbers) on them like this: https://i.stack.imgur.com/ET4ny.jpg which you can use to represent how far the link is, or how costly it is to go that way (fuel cost, or travel time, or effort, or speed limit on the roads, or bus/train/plane ticket price) and then you can ask the cheapest way to visit all the places, or the shortest way, or the fastest way. So it can be used in route planning (I want to fly here to here, via somewhere, what are my options?)

Because it's about connections, not location or shape, it's very general. It can talk about computer networks like this: https://static.packt-cdn.com/products/9781788621434/graphics... and you can see one choke point in the middle that has to be fast enough to take the aggregate traffic of all the computers on both ends. Or you can look at it for the reliability - that single middle link is a good place to make two links, because then one can fail and all the computers are still working.

Then you can deal with different "shapes" of graph (not layout on paper, "shape" of connections): http://2.bp.blogspot.com/-GW8bGXZNrWg/VmFGCI949QI/AAAAAAAACd... does each node connect to every other one? Is it sparsely or densely connected? How many links could we lose and leave the minimum spanning tree - the skeleton network where everything is still connected end to end by one link? Is there one critical link which would separate it into two disconnected parts? What's the worst case for any two nodes? What if one link fails, what happens to the best and worst cases?

It can describe "shapes" of communication or organisation - military has a top-down structure, anarchy has a meshed everyone-to-everyone structure.

Graphs can be directed, edges can be one-way, they can be used in project planning, nodes can be tasks and edges can be which task output feeds into the next task input, and tasks and edges can weight how long things take: https://2.bp.blogspot.com/-SHnStluEIPc/WkarHINi08I/AAAAAAAAQ... then you can ask what things you can arrange to do at the same time and what you can't. In the picture there's a 3 day task waiting on a 4 day task. No matter how quickly you do all the other tasks, the whole thing must take 7 days minimum. This comes round to computing and how quickly you can speed up a program by adding multiple-processors. If there's a chain like that, only speeding up that chain can help, nothing else can help.

State transition diagrams are graphs: https://faculty.etsu.edu/tarnoff/ntes2150/statemac/states1.g...

They come into computing, tree structures are connection graphs, regular expressions are state transition graphs, concurrent programming is about tasks you can do at the same time, the internet is a connection graph and routing is finding short paths between distant computers through other systems.

Neural Networks are about connection graphs - each node is a neuron holding an activation value and when it triggers, it sends some activation out to the neurons it's connected to. If the combined input passes that neuron's activation value, it does the same. Somehow by adjusting these trigger values and feeding a prepared input in (pixel values from a photo, one value to each input neuron) it triggers a cascade of activation through the whole network, and it settles on an output high for a picture of a dog, low for anything else.

And concept maps, knowledge graphs, can be modelled like this; which ideas are connected to other ideas? When learning something it can help to make dense connections - instead of trying to remember that "shoe" is "zapato" in Spanish as a plain word connection which will be easy to forget, try and have it in a sentence about how your shoes are pinching your feet, and one about the smell of leather shoes, and one about the slimy feel of shoe polish, and a visual memory of the nearest shoe shop. More dense connections give you more ways to access that memory, more redundant, more easily, and using the mental connections reinforces them.

Note taking tools like Obsidian, Dendron, TiddlyWiki, and systems like Zettelkasten are working with the problem "when I've taken notes, I can never find them, and hardly use them", and saying you need to connect the notes to other notes, more connections, then you see one and it gives you ideas by seeing what it links to - last time you used this note, what else were you thinking about?

Wikis are graphs, HyperText (web) links make a spider's web of connections between pages.

This is the "graph" in Facebook's "social graph" - who knows each other, how do they know each other, how strong are the connections between people? You know one person as a coworker, another by being in a hobby group, another is an extended family member and a close friend, another your phones both see the same WiFi access points so you must live or work near each other.

It's so general it comes up all over the place; how do decisions get through your company from the people who make them to the people who need to hear them? How does Google Maps find you a good route? How do you deliver post around the country moving it from regional post office to central sorting hub back to regional delivery office? How does an AI path-find a route in a computer game? Which routes do you send trucks and cargo ships so they avoid making a return journey carrying no cargo, or never go via a bridge they can't go under? How do you build a country-wide telephone network without bankrupting yourself trying to run a copper wire from every person to every person? How do you represent the connections in your supply chain from company to company so you can avoid a 'chip shortage' event and have redundant suppliers if one of them has problems? Where does the water in the heating system need to go to get to all the radiators? Who is only six degrees from Kevin Bacon, where people are connected by appearing in the same film as each other? Who has the lowest Erdos number, where people are connected by being named in the same math papers as each other? If someone watches a VSauce YouTube video, which channels might they be interested in being recommended?

The study of "stuff which is connected". ok I will link Wikipedia https://en.wikipedia.org/wiki/Graph_theory#Applications