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Graph theory is a branch of mathematics that studies networks of connected nodes and edges. A graph is made up of vertices (nodes) and edges (lines connecting the vertices). Understanding graph theory helps us model and solve problems about various types of networks, such as social networks, computer networks, and transport systems.
A key concept in graph theory is the distinction between different types of paths and cycles. An **Eulerian circuit** is one such cycle and it specifically requires you to travel through every edge exactly once and return to the starting vertex.
Graph theory uses specific definitions and rules to understand different structures and solve complex problems efficiently.

A connected graph is one in which there is a path between every pair of vertices. This means you can travel from any vertex to any other vertex within the graph without lifting your pen off the page.
For a graph to have an Eulerian circuit, it must be a connected graph. A disconnected graph would have at least one pair of vertices with no path between them, making it impossible to form a single continuous cycle that covers every edge.
Real-world examples of connected graphs include road networks within a city, where you can drive between any two points, provided there are no closed roads.

The degree of a vertex in a graph is the number of edges that connect to it. In other words, it’s the count of how many connections a particular node has.
An **important criterion** for the existence of an Eulerian circuit is that every vertex in the graph must have an even degree. This ensures that if you start and finish at the same vertex, you can enter and exit every vertex the same number of times.
To illustrate: if one vertex in the graph has an odd degree, you would either be stranded at that vertex or couldn't return to the starting point, breaking the criteria for an Eulerian circuit.
So, remember: for an Eulerian circuit, all vertices must hook up with an even number of edges.