example we can study air transportation by constructing graph gcalled flight networkwhose vertices are associated with airportsand whose edges are associated with flights (see figure in graph gthe edges are directed because given flight has specific travel direction the endpoints of an edge in correspond respectively to the origin and destination of the flight corresponding to two airports are adjacent in if there is flight that flies between themand an edge is incident to vertex in if the flight for flies to or from the airport for the outgoing edges of vertex correspond to the outbound flights from ' airportand the incoming edges correspond to the inbound flights to ' airport finallythe in-degree of vertex of corresponds to the number of inbound flights to ' airportand the out-degree of vertex in corresponds to the number of outbound flights lax aa dfw dl aa aa jfk dl ua ua ord sfo bos nw sw aa aa mia figure example of directed graph representing flight network the endpoints of edge ua are lax and ordhencelax and ord are adjacent the in-degree of dfw is and the out-degree of dfw is the definition of graph refers to the group of edges as collectionnot setthus allowing two undirected edges to have the same end verticesand for two directed edges to have the same origin and the same destination such edges are called parallel edges or multiple edges flight network can contain parallel edges (example )such that multiple edges between the same pair of vertices could indicate different flights operating on the same route at different times of the day another special type of edge is one that connects vertex to itself namelywe say that an edge (undirected or directedis self-loop if its two endpoints coincide self-loop may occur in graph associated with city map (example )where it would correspond to "circle( curving street that returns to its starting pointwith few exceptionsgraphs do not have parallel edges or self-loops such graphs are said to be simple thuswe can usually say that the edges of simple graph are set of vertex pairs (and not just collectionthroughout this we assume that graph is simple unless otherwise specified