### Summary

Graph Theory forms the foundation of various AI and ML models. The theory is about mathematical structures called graphs, constructed to represent pairwise relationships between objects, allowing for efficient problem-solving in AI.

### ELI5

Imagine you’re playing a game of connect-the-dots, where you draw lines (edges) to join together certain dots (nodes). The game you’re playing is actually a very simple example of graph theory. In AI, these dots and lines can symbolise and show relationships between different pieces of information.

### In-depth explanation

Graph Theory is a branch of mathematics that studies graphs, which are mathematical structures used to model relationships between objects. A graph consists of nodes (also called vertices) and edges (also called arcs or lines) which connect these nodes. In the context of AI and ML, Graph Theory is crucial, especially in formulating solutions to computational problems and modeling networks.

The fundamental element in Graph Theory is the concept of a graph itself. A graph is defined as a pair of sets - one set of vertices and one set of edges, where each edge connects a pair of vertices. Graphs can be directed or undirected, weighted or unweighted. Directed graphs include a direction with each edge, while undirected graphs do not. In a weighted graph, each edge has a weight (or cost), while in an unweighted graph all edges have uniform weight.

Graph Theory is widely used for solving various types of problems in computer science. For instance, social networks can be represented as graphs where nodes represent users and edges represent the relationships between them. Similarly, in transportation logistics, cities can be represented as nodes and roads as edges, with the weights of these edges representing the cost of travel.

In AI and ML, Graph Theory serves as the basis of graph-based algorithms such as Graph Neural Networks (GNNs) that operate on graph data structure. Essentially, with graph structures, data points can have complex and irregular relationships with one another, compared to traditional ML algorithms that often assume that data points are independently and identically distributed.

Another crucial application of Graph Theory in AI is Graph Search Algorithms. Algorithms like Breadth-First Search, Depth-First Search, A* Search, Uniform Cost Search, etc., utilize Graph Theory to efficiently search through large data structures or solve optimization problems. These algorithms are often employed in fields such as Path Planning in Robotics, AI in Games, Network Routing, etc.

Finally, Graph Theory is also leveraged in computational biology, physics, and complex network research due to its ability to represent complex structures and relations effectively.

### Related terms

Graph Neural Networks, Graph Search Algorithms, Deep Learning, Path Planning, Network Routing, Optimization, Computational Biology, Social Networks, Graphical Models