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.
In addition, there are a few different types of graphs including Eulerian graphs and Hamiltonian graphs. A graph is Eulerian if it is connected and \(G\) contains a closed trial with the use of every edge exactly once. Within Eulerian graphs, one can begin and end with any vertex. A graph is Hamiltonian if it contains a cycle that goes through every vertex.