 In many cases we want to represent the existence of a relation between two objects. For example, two cities might have a direct flight between them. Two people might be related. Two islands might have a bridge between them. We can represent these relationships using a graph. Now before we continue, it's worth keeping in mind that the more names we have for something, the more important it probably is. So, while we often use the term graph, this is also called a network, and it's also known as a web, unless it's called a sociogram. But to be consistent, we'll call it a graph, except when we don't. Informally, a graph consists of points, some pairs of which are joined by lines. Or maybe they're called vertices and they're joined by edges. Or maybe they're called nodes and they're joined by links. Since different fields use different terms, we won't make any attempt to be consistent. A graph is a web, is a network. A point is a vertex, is a node. A line is an edge, is a link. Now just as you can tell something about a person by what they call a carbonated drink, you can also tell something about a person by which terms they use. Generally, mathematicians use a term graph, vertex, and edge. But computer scientists usually use a term network, node, and link. Now if a graph represents relationships, these relationships could be directional. Links go from one web page to another, but not necessarily in the other direction. Or these relationships could be weighted. It might cost more to fly from Boston to Providence than from Boston to London. Or they could loop. A person could be their own grandfather. And there could be multiple connections. Two islands might have several ridges joining them. Likewise, a node might be weighted. Some friends might be more influential than others. While all of these modifications could make the study of graphs much more useful, for now we'll assume that nodes and edges are unweighted, edges are bi-directional, no edge connects a node to itself, and two nodes are connected with that most one edge. This gives us a simple graph. It's worth mentioning that while a simple graph does prohibit two nodes being connected with both in one edge, a lot of the graphs we study will have these multiply connected nodes. In fact, the graph from the Bridges of Kennecksburg problem is one of these graphs that have multiply connected nodes, but we still include it in elementary graph theory. Formally, a graph G is a tuple, v e, where v is a non-empty set of points, vertices, nodes, and e is a possibly empty set of lines, edges, links, which consist of sets of two vertices. So we might try to describe the graph shown. We see there are four vertices, one, two, three, and four, and the edges are described by the vertices they join. So we have an edge between vertex one and three, two and three, two and four, and three and four. And since the edges are sets, the order in which we list the vertices doesn't matter. So the edge we described as one and three is the same as the edge we described as three, one, or we could be given the set of vertices and the set of edges. And so there are four vertices. And remember, if it's all written down, it didn't happen. Label everything. So we'll put down our vertices, and it really doesn't matter where we put them. Our list of edges indicates that we have an edge joining vertices one and two, so we'll draw a line between them. We also have an edge between one and three, and one and four. So we'll draw those in as well. Given an edge, we say that e is incident on vertices v one and v two, and that the two vertices are adjacent. Informally, any edge is incident on the vertices corresponding to its endpoints. So in the graph shown, the edge e one is incident on the nodes one and two, and the edges e two, e three, e four are all incident on vertex three. It's important to understand that in the representation of the graph, only the identified points and lines exist. Since the graph is the relationship between the edges and their incident points, if two edges cross, they do not create a point. And in fact, it is irrelevant where points are with respect to each other. And what this means is that we can move points around. So we can move point a over and point e over. And since we haven't changed the relationships between the points and the lines, it's the same graph.