**In this post, we delve into the world of mathematical optimization within graphs, exploring key concepts, algorithms, and practical applications. Graph problems can be found in many places. Obvious ones are in logistics or social network analysis, like finding the optimal route for a delivery company or the lowest amount of connections between two people. But did you know that graphs are also applicable in urban planning, disease transmission modeling, fraud detection, recommendation engines, and cybersecurity? By leveraging optimization algorithms specifically designed for graphs, data scientists can uncover optimal solutions, allocate resources efficiently, and make data-driven decisions.**

First, we’ll start with an introduction section, to explain the basics of graphs. Then we dive into common graph problems and algorithms trying to solve these problems.

As a recap, below the basics on graph theory.

## What is a graph?

A graph consists of vertices (or nodes) and edges. If the vertices are related in a certain way, they are connected with an edge. To define a graph, you need the names of all the vertices and you need to know which vertices are connected.

Below a graph that has vertices {A, B, C, D, E} and edges {{A, D}, {A, E}, {B, C}, {B, D}, {C, D}}.

Sometimes, graphs can contain *loops*. A loop is an edge that has the same start and end node (a node is connected with itself).

Other terms that are nice to know in graph theory:

- The
*order*of a graph is equal to its number of vertices. - The
*size*of a graph is the number of edges (sometimes plus the number of vertices). - The
*degree*of a vertex is the amount of edges it has (a loop is counted twice for the beginning and end point).

## Common variations

The previous graph example is also called a *simple graph*, because it only contains vertices and (undirected) edges. But you can easily make it a bit more complex, and often…