"What I love about the topic of graphs is just using a handful of different algorithms, you can cover the majority of graph problems."
"There are a whole class of problems that can be represented by graphs, encoding very rich structural information."
"It's like a complete bipartite graph, and you're passing messages between nodes."
"The goal of this algorithm is to find the shortest path from point A to point B in a graph."
"Now the user can type in all their vertices and all their edges."
"Traversal algorithms are algorithms used to traverse or visit nodes in a graph."
"Graphs naturally generalize objects such as grids or sequences, where machine learning has already made very significant strides."
"When does the graph b to the x lie tangent to the line y equals x?"
"Understanding how a graph is laid out, how it looks in code, and just in general how you can use them to do different calculations is super crucial."
"Dijkstra's algorithm works for graphs with cycles, but all edge weights have to be either 0 or positive."
"A walk in a graph is a sequence of vertices and edges."
"But the neat thing about graph theory is that it also find ways to show up in the most unexpected places."
"Let's now proceed to formally define a graph."
"So every vertex could be its very own group."
"This is the causal connection graph and this causal connection graph is saying you can go from 1 to 8 but you can't get back from 8 to 1."
"You can see a category as the transitive closure of a graph."
"I suspect there is a group theoretic description because it's a vertex-transitive graph."
"Topological ordering is a linear ordering of a graph's nodes where every directed Edge UV blah blah blah."
"Once we've reached the depth for that DFS we've actually just traversed a mini topological ordering."
"Try the function 1 over X. Its graph forms a vertical asymptote at x equals zero, meaning the height of the graph grows without bound as X approaches zero."
"Understanding these complex networks through graph theory is something that has become more and more popular."
"Taking from graph theory, branching it up to the connectome and understanding the actual networks of the brain allow us to understand these behavioral abnormalities."
"Graph theory is all about relationships."
"Every map can be colored using four colors, such that two neighboring countries are different colors."
"Thinking about navigation through systems actually gives us a little bit of history because, in fact, the very first reported use of graph theory was exactly this problem."
"This knows automatically what to do once you build the graph. Very, very nice."
"Graphs allow you to detect implicit relationships in your factual knowledge."
"Graph data science... is when we're using the relationships in our data in addition to just sort of discrete data points to answer questions."
"Community detection... we're interested in finding clusters or partitions in the graph."
"Graph theory is a rich toolkit of things that you can do to analyze your data and give you intelligent looking answers from it."
"What we want to do as decision machine learners is to find a good graph to describe that big bunch of points using less details, maybe less parameters."
"Community detection is a very useful technique in graph analytics."
"I learned graph theory in computer science before I learned that it was a sociological concept."
"A state machine can be seen as a directed graph."
"If all of the eigenvalues of a graph have bounded multiplicity, you can test isomorphism in polynomial time."
"When I look at that beautiful picture, I know everything about this graph."
"A nice drawing would reveal structure, and a random graph should not have that structure."
"If you have a nice picture of a graph, you can show that there will be a large set of vertices with small boundary."
"This graph, although it has a discontinuity at zero, happens to sum up to a finite number; it converges."
"A tree is a directed graph and in particular, it has no cycles."
"People in the network science literature often measure things like the radius of a graph and some other kind of measures that are trying to tell you something about like is a graph a long spread out thing like a line graph or something super compact like a star."
"So in this problem, we're given an undirected graph, and as usual, we will call him G."
"If we want to learn a property not of a vertex but of the entire graph, one thing we can do is to find the radius."
"This problem is screaming out graph theory."
"We're going to use a sort of a canonical trick in graph theory."
"Vertex cover is the problem where we're given a graph G and an integer k, and we want to know whether G has a vertex cover of size at most k."
"Because our graph is acyclic, we know it has a topological order."
"The number of vertices in my graph here is big O of r."
"The depth first search algorithm can dig around in a graph and tell me all of the things that are reachable from a given source."
"The more technical term for a big connected clump is a connected component."
"We can use depth first search to solve this problem pretty easily."
"So let's talk about the absolutely fundamental problem of searching a graph."
"Overall, there's many cool unsolved questions within graph theory, and this is cool because it's one of the simpler to explain ones."
"It's about creating patterns in a graph, patterns of nodes and relationships, and then it's about finding those patterns when you're doing your queries."
"You could also take a set of ordered pairs and put it together onto a graph, and it gives us this beautiful visual picture."
"If we want to preserve the tree structure, what we need is that every node has a unique parent and that the tree contains no cycles."
"The good news about graph theory is the definitions are very short and precise, and there is no ambiguity in understanding them."
"The eccentricity of a vertex is the maximum distance between this vertex and any other vertex on the graph."
"Graph diameter is just the largest eccentricity, which is the largest possible distance between any two nodes in the graph."
"The central point in the graph is the point where the eccentricity of that point is equal to the graph radius."
"Graph Center is a set of all central nodes, or a set of vertices whose eccentricity is equal to the radius."
"The definition is very clean and it's very clear which nodes are called center and which are called periphery."
"The major contribution of the PageRank algorithm was in the fact that how to handle directed graphs."
"Graph convolution computes new feature vectors for each vertex, propagating information along the edges of the mesh."
"The total distance traveled is the area under the graph."
"Graph diameter is the largest shortest path."
"The transitivity of the graph is measured as three times the number of triangles divided by the number of connected triples."
"If your degree is five, you can change directions at most four times."
"Your end behavior is determined by your leading terms."
"You can represent all these problems as graphs."
"The graph is defined by three entities: the set of vertices, the set of edges, and the adjacency matrix."
"The Laplacian is nothing else than a measure of smoothness of a function on a graph."
"The Boost Graph Library is actually pretty simple to use."
"There are three things that you need to think about: how to store your graph structure, how to associate properties with your edges and vertices, and how to get those properties in and out of an algorithm."
"Computational graphs are an abstract way of describing computation as directed graphs."
"Each node in this directed graph is going to represent a random variable."
"Each node in the graph corresponds to a random variable and has a conditional probability table associated with it."
"A continuous loop will always have the same number of nodes and the same number of links, and that will always be an even number."
"Points representing objects or data are called nodes, while the predicates that connect them are called edges."
"The things to have in your head here are graphically what's going on to the base set as we're deleting nodes from it or removing variables from compound factors that we've built up."
"There exists graphs for which there exists a good elimination ordering, but finding that elimination ordering is itself an NP-hard problem."
"A Hamiltonian cycle in a directed graph is a simple cycle that visits every vertex exactly once."
"The job of the verifier is just to check that that really is a Hamiltonian cycle in the graph."
"Make edge weights non-negative while preserving shortest paths."
"A strongly connected component in a directed graph is a component such that all the vertices in that component are reachable from every other vertex."
"A graph is a set of a bunch of nodes and edges connecting the nodes."
"Dijkstra's can tell you the shortest path between that node and all other nodes in your graph, which is pretty sweet."
"Scale-free graphs are most common in nature and they have very special properties."
"Among all paths between two vertices, it's going to be one with the minimum weight."
"We're repeatedly going to lower that value as we gain more information about the graph."
"So here, in this graph, there is a Hamiltonian path that visits every vertex exactly once."
"Think about the edges as the pipes and the nodes or the vertices in the graph as the pipe joints."
"As the water flows from higher altitude to lower altitude, similar concept is applied here; each vertex is associated with a height."
"Excess flow is nothing but whatever amount of the flow it is receiving into one node and whatever amount of the flow is going out of the node."
"The algorithm is very simple: you just initialize the graph once and keep doing the push and relabel operation until there is no more push and relabel operation left."
"A relabel operation is nothing but when the height of a vertex is low and the vertex is overflowing, then you adjust the height to make it higher than its neighbor vertex."
"The time complexity turns out to be in the order of V squared multiplied by E, where V is the number of vertices and E is the number of edges in the graph."
"A few words just sort of reminders from graph theory."
"Neo4j and graph databases are the best when you're working with data that's related to each other."
"A Markov random field is defined like this: we are given a graph and what we call clique potentials, which replace the conditional probability distributions that we have for a Bayesian network."
"With the right amount of nodes... it's a pleasing way, it's a pretty way to draw a graph."
"Graph-based recommendations have transformed the consumer landscape."
"If by looking at that graph you can understand properties of the distribution they specified, that's nice because that's much easier to do."
"When you're trying to optimize your graph algorithm, you should definitely test it on different types of graphs and see where it works well and where it doesn't."
"The period of a sine or cosine graph is equal to 2 pi divided by K."
"If we draw a vertical line through the graph and the vertical line only touches the graph at one place no matter where we draw it, then it's a function."
"I want to talk about the fascinating topic of strongly connected components."
"Strongly connected components... I like to think of them as self-contained cycles within a directed graph."
"Having mastered computing the connected components of undirected graph in linear time, let's now turn our attention to directed graphs."
"The graph is strongly connected if you can get from any one point to any other point, and vice versa."
"It's not even so obvious how to define pieces, how to define connected components in a directed graph."
"Polynomial function graphs are continuous and smooth."
"The graphs are parabola; it is up if a is positive, it is down if a is negative."
"Homogeneous graphs are graphs in which the nodes of the graph belong to one single type."
"Once we have found an active path, we know that just based on the graph structure, we cannot guarantee the independence."
"If the middle node is observed, we have an active triple."
"An active triple makes the entire path active."
"Once we find an inactive triple along a path, the entire path is inactive."
"There is an efficient algorithm for finding the shortest route between two vertices."
"Dijkstra's algorithm can be used to construct a spanning tree T with the following property: for any vertex Y in the graph, the unique XY path in T is actually a shortest L shortest XY path."
"The importance of this cannot be overstated. It makes it possible to handle bipartite graphs of a ridiculous size."
"An Eulerian circuit is an Eulerian path which starts and ends on the same vertex."
"The optimal way to represent our graph is with a two-dimensional adjacency matrix."
"If a graph has an Eulerian circuit, it also has an Eulerian path."
"A tree is simply an undirected graph with no cycles."
"Directed acyclic graphs... often represent structures with dependencies."
"A topological ordering of nodes is an ordering of nodes that tells you how to process the nodes of the graph so you don't perform a task before first having completed all its dependencies."
"A minimum spanning tree is a subset of the edges that connects all the vertices together without any cycles and with minimal possible total edge weight."
"A depth first search plunges depth first into a graph without regard for which edge it selects next, until it cannot go any further at which point it backtracks and continues its exploration."
"Dijkstra's algorithm is a single source shortest path algorithm for graphs."
"The one main constraint for Dijkstra is that all edges of the graph need to have a non-negative edge weight."
"In today's Wrath of Math lesson, we'll be proving Euler's formula for connected plane graphs."
"Every tree graph has one less edge than it has vertices."
"The number of regions is one because you would need a cycle in order to get more regions."
"N minus M plus R is equal to two."
"PageRank basically measures the importance of each vertex in a graph."
"For example, the problem of finding the shortest path between two nodes in a graph is in P, there's a polynomial time algorithm for it."
"The following matrix will show the shortest distances between every pair of vertices."
"In a bipartite graph with partite sets U and W, where U is the smaller partite set, there exists a matching that covers the smaller partite set if and only if that graph satisfies Hall's condition."
"A cycle is a path in the graph such that the first and last vertex is the same."
"A strongly connected component is a subset of vertices in a graph such that all the vertices are reachable from each other."
"If we can find a matching that covers every vertex in the smaller partite set, then the graph satisfies Hall's condition."
"So what exactly will we cover in this video? First, we'll define the term graph, we'll learn some key terms such as vertices and edges."
"A tree is really just a special kind of graph with a couple different parameters that must be true for something to be a tree versus just a regular old graph."
"The adjacency matrix is a pretty interesting one, it's sometimes hard to wrap your head around at first, but once you understand it, it's a very helpful way to visualize the graph."
"There only exists a single path that's unique between any two nodes."
"The graph of an equation is the set of points \( (X, Y) \) that satisfy the equation."
"A complete graph reveals all the important features of the graph."
"So now I've colored every node in G, the n plus 1 node graph, safely using a total of d plus 1 colors."
"The graph of \( f(x) = x^4 \) will take a 'U' shape, but notice that at the vertex we have a flat bottom, which makes it unique in comparison to the quadratic graph."
"We're going to take it to new heights of graph knowledge, and you'll be able to do many more cool things that you're able to do before."
"So the total sum of out degrees equals the total sum of in degrees."
"The earliest graph theory problem, for sure, Known to human civilization."
"If you have strong N consistency where you have N nodes in your graph, you can solve without backtracking."
"So if the graph has a cycle, you can't solve the problem. It has no solution."
"So topological sorts only works on acyclic graphs."
"Needs to be acyclic, otherwise there's no solution."
"A graph with no cycles is called a DAG, so directed acyclic graph."
"Articulation point is a vertex in connected undirected graph such that removing that vertex will disconnect the graph."
"If there's a back edge, it means there's another path to connect two vertices so the current vertex cannot be an articulation point."
"A vertex is an articulation point if it satisfies any of these two conditions: one is if it's a root vertex with two independent children."
"We're proving that for a planar graph, its number of edges is less than or equal to three times its number of vertices minus six."
"If a graph is planar and connected, then its number of edges can at most be three times its number of vertices minus six."
"Find the minimum cost spanning tree or minimum spanning tree given a graph with positive weights."
"Clustering coefficient is all about trying to understand how connected are the neighbors of a given node."
"This is an important graph, and it's closely connected to Laplace's equation."
"Using graphs to model complex networks is great."
"The big O runtime for depth first and breadth first search is all of vertices plus edges."
"Finding the shortest distance between vertices is one of the most common applications of graph searches."
"The node degree is defined to the number of nearest neighbors."
"The key idea that GMATA uses to solve the few-shot learning problem on graphs is that it will specify a signature function for each learning task."
"The very first model is actually a random graph model that was proposed in 1959."
"If discovery time of a node u is less than or equal to low time of a child v, then u is an articulation point."
"A single root has to pass through that x-axis."
"D's algorithm determines the shortest paths between nodes in a graph."
"It has a unique graph; it's not a family of graphs but rather it is one of the members of the family of curves of the general solution."
"The lightest weight edge coming out of that vertex is in the minimum spanning tree."
"The output is a spanning tree, T, and by spanning tree, we mean it connects all the vertices."
"We want to generate graphs via sequential adding of nodes and edges."
"Given the graph, given the actions we have made so far, conditioning on that, what is the next action we want to take?"
"A graph plus a node ordering gives us a unique sequence of sequences to generate a given graph."
"The sequence is actually a two-level sequence. There is a sequence of adding nodes, and there is a sequence of adding edges."
"A connected and acyclic graph is called a tree."
"The Laplacian matrix is a generalization of the Laplacian operator onto the world of graphs."
"Drawing graphs of functions tells you about the very nature of functions."
"Horizontal asymptote, end behavior, x approaching infinity - they're all equivalent."
"The key ingredient in making this scientific breakthrough was the use of graph representation and the graph neural network technology."
"Antibiotics are small molecular graphs and we can represent molecules as graphs where the nodes are atoms and edges correspond to chemical bonds."
"Graph theory provides a very visual way to analyze problems from almost any field of mathematics."
"The shortest path problem is probably one of the most important questions in graph theory."
"The right algorithm is to take the very smallest edge and keep adding edges smallest at a time, making sure you never get a cycle."
"The first step is to compute all distances to the vertex."
"The graph is fully parametric and we can make changes to the path anytime we'd like."
"Graphs model different types of networks, and there are a number of blazingly fast primitives for reasoning about their structure."
"Breadth-first search essentially means you are traversing outwards towards your subsequent neighbors first before going out to their neighbors."
"The graph lets you build in on every natural human advantage you have."
"Graph algorithms are really cool because they force you to really distinguish between some intuitive idea you have about how to do something and actually making that implementation work."
"If the boolean formula can be made true, then the answer to the three color problem will be yes."
"To colorability is really an easy problem; it's the same as checking whether our graph is a bipartite graph."
"Every single graph that has no cycles has a kernel."