On Eulerian and Hamiltonian Graphs and Line Graphs

Source abstract

A graph G has a finite set V of points and a set X of lines each of which joins two distinct points (called its end-points), and no two lines join the same pair of points. A graph with one point and no line is trivial. A line is incident with each of its end-points. Two points are adjacent if they are joined by a line. The degree of a point is the number of lines incident with it. The line-graph L(G) of G has X as its set of points and two elements x, y of X are adjacent in L(G) whenever the lines x and y of G have a common end-point. A walk in G is an alternating sequence v 1 , x 1 , v 2 , x 2 , …, v n of points and lines, the first and last terms being points, such that x i is the line joining v i to v i+1 for i=1, …, n-1.