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A graph is an ordered pair $G = (V, E)$ comprising a set $V$ of vertices or nodes together with a set $E$ of edges or arcs, which are 2-element subsets of $V$.

A subgraph is a graph $S = (V', E')$ where $V' \subseteq V$, $E' \subseteq E$, and every edge in $E'$ connects two vertices in $V'$.

The degree of a vertex in a graph is the number of edges that are incident to the vertex, with loops counted twice.

A path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another.

A graph is connected if there is a path between every pair of vertices.

A cycle in a graph is a path that starts and ends at the same vertex and includes at least one other vertex, with no vertices repeated.

A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge.

An adjacency matrix is a square matrix used to represent a finite graph where each element of the matrix indicates whether pairs of vertices are adjacent or not in the graph.

An incidence matrix is a matrix that shows the relationship between edges and vertices in a graph where rows represent vertices and columns represent edges, indicating which vertex is incident to which edge.

A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets $U$ and $V$ such that every edge connects a vertex in $U$ to one in $V$.

A planar graph is a graph that can be drawn on a plane without any edges crossing each other.

Euler's formula states that for a connected planar graph with $V$ vertices, $E$ edges, and $F$ faces (including the outer one), the formula $V - E + F = 2$ holds.

A Hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once.

A Hamiltonian cycle is a cycle in an undirected or directed graph that visits each vertex exactly once and returns to the starting vertex.

A tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, a connected graph without cycles.

A forest is a disjoint union of trees, or equivalently, an undirected graph without cycles.

An Eulerian path is a path in a graph which visits every edge exactly once.

An Eulerian cycle is a cycle in a graph that visits every edge exactly once and returns to the starting vertex.

An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a vertex in the graph.

The chromatic number of a graph is the smallest number of colors needed to color the vertices of a graph so that no two adjacent vertices share the same color.

A directed graph or digraph is a graph in which the edges have a direction, indicated by an arrowhead, and the edges are called arcs.

A weighted graph is a graph in which each edge is assigned a weight or cost, which is often used to represent the cost of traveling between the vertices connected by the edge.

A simple graph is a graph that does not contain any loops or multiple edges. The edges in a simple graph are implied to be undirected.

The Handshaking lemma is a theorem in graph theory that states that, in any undirected graph, the sum of the degrees of all vertices is equal to twice the number of edges.