We will use that convention here. If xy is an edge in a graph G, We say that x and y are incident with the edge and we say that y is adjacent to x. For an undirected graph, x is adjacent to y if and only if y is adjacent to x. The degree of a vertex is the number of edges incident with it. We say the path has length n.
G is acyclic if G contains no cycles.
G is a maximal connected subgraph of G. Exercise 19.1. It turns out that if a graph G is connected and has exactly two vertices of odd degree then G has a euclidean cycle. A tree is a connected acyclic graph. For any tree we can choose one vertex and designate it as the root . Once we have done this, we can talk about the parent of a vertex or the children of a vertex, or the depth of a vertex, in relation to the root we have chosen.