Summary of Chapter 2

1 A graph consists of vertices(nodes) which are connected by edges(arcs). 2 A subgraph is part of a graph. 3 If a graph has a number associated with each edge (its weight) then the graph is a weighted graph (network). 4 The degree or valency (or order) of a vertex is the number of edges incident to it. 5 A path is a finite sequence of edges, such that the end vertex of one edge in the sequence is the start vertex of the next, and in which no vertex appears more than once. 6 A walk is a path in which you are permitted to return to vertices more than once. 7 A cycle (circuit) is a closed ‘path’, i.e. the end vertex of the last edge is the start vertex of the first edge. 8 Two vertices are connected if there is a path between them. A graph is connected if all its vertices are connected. 9 A loop is an edge that starts and finishes at the same vertex. 10 A simple graph is one which there are no loops and not more than one edge connecting any pair of vertices 11 If the edges of a graph have a direction associated with them they are known as a directed edges and the graph is known as a digraph. 12 A tree is a connected graph with no cycles. 13 A spanning tree of a graph, G, is a subgraph which includes all the vertices of G and is also a tree. 14 A bipartite graph consists of two sets of vertices, X and Y. The edges only join vertices in X to vertices in Y, not vertices within a set. 15 A complete graph is a graph in which every vertex is directly connected by an edge to each of the other vertices. If the graph has n vertices the connected graph is denoted kn. 16 A complete bipartite graph (denoted kr,s) is a graph in which there are r vertices in set X and s vertices in set Y. 17 Isomorphic graphs show the same information but are drawn differently. 18 An adjacency matrix records the number of direct links between vertices. 19 A distance matrix records the weights on the edges. Where there is no weight, this is indicated by ‘-‘.

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