# Graph Theory

Graph theory is a topic used in discrete mathematics to show networks and study relationships between objects in a more mathematical way. Graphs consists of a a set of vertices usually denoted $$V,$$ and an sets of edges typically denoted $$E.$$ Each edge in a graph connects the vertices. A graph $$G$$ is defined as an ordered pair where $$G=\left(V,E\right).$$
The vertices in the graph are $$V=\left\{1,2,3,4,5,6\right\}.$$
The edges in the graph are $$E=\left\{\left\{1,2\right\},\left\{1,5\right\},\left\{2,3\right\},\left\{2,5\right\},\left\{3,4\right\},\left\{4,5\right\},\left\{4,6\right\}\right\}$$.
There are a few concepts within graph theory that include walks, paths, trails and cycles. A walk on a graph is defined as a sequence of adjacent vertices where repetition is allowed. A path is a walk, however, no vertices can be repeated in this case. Notice that within these two concepts, it is known that if a walk exists between $$x$$ and $$y$$, then a path also exists from $$x$$ to $$y.$$ Next, a trial is defined as a walk that has no repeated edges and a trail may be closed if it starts and ends with the same vertex. A cycle is a closed trail that does not have any repeated vertices besides their endpoints.