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Theorem 15.2 The number of edges in the complete graph on n vertices is n(n 1)/2. There are n vertices, and each of them could be adjacent to n 1 other vertices. So there are n(n 1) ordered pairs of vertices. Therefore, there are n(n 1)/2 unordered pairs because each unordered pair could be ordered in two ways. For example, the unordered pair {a, b} can be ordered as either (a, b) or (b, a). For example, the number of edges in the complete graph on the four-vertex set in Example 15.3 is n(n 1)/2 = 4(4 1)/2 = 6. Corollary 15.1 The number of edges in any graph on n vertices is m PATHS AND CYCLES A walk from vertex a to vertex b in a graph is a sequence of edges (a0 a1, a1 a2, . . . , ak 1 ak) where a0 = a and ak = b, that is, a sequence of edges (e1, e2, . . . , ek) where if edge ei connects some vertex to vertex ai then the next edge ei+1 connects that vertex ai to some other vertex, thereby forming a chain of connected vertices from a to b. The length of a walk is the number k of edges that form the walk. n(n 1)/2.

CHAP. 15]

Although a walk is a sequence of edges, it naturally induces a sequence of adjacent vertices which the edges connect. So we may denote the walk (a0 a1, a1 a2, . . . , ak 1 ak) more simply by a0 a1 a2 . . . ak 1 ak as long as each pair (ai 1, ai) is a valid edge in the graph. If p = a0 a1 a2 . . . ak 1 ak is a walk in a graph, then we refer to p as a walk from a0 to ak (or from ak to a0), and we say that p connects a0 to ak and that a0 and ak are connected by p. We also refer to a0 and ak as the terminal points or the end points of the walk. A path is a walk whose vertices are all distinct. EXAMPLE 15.4 Graph Paths

In the graph in Figure 15.4, abcfde is a path of length 5. It is, more formally, the path (ab, bc, cf, fd, de). The walk abefdbc of length 6 is not a path because vertex b appears twice. The walk abefa of length 4 is also not a path. The sequence abf is not a walk because bf is not an edge. And the sequence abb is not a walk because bb is not an edge. Finally, aba is a walk of length 2, and ab is a walk of length 1.

A graph is said to be connected if every pair of its vertices are connected by some path. A graph that is not connected is called disconnected. All the graphs in the previous examples are connected. EXAMPLE 15.5 A Disconnected Graph

In this chapter we concentrate on the rm s production costs or what lies behind its supply curve. Explicit costs are the actual, out-of-pocket expenditures of the rm to purchase the services of the factors of production it needs. Implicit costs are the costs of the factors owned by the rm and used in its own production processes. These costs should be estimated from what these factors could earn in their best alternative use or employment. In economics, costs include both explicit and implicit costs. Pro t is the excess of revenues over these costs. Example 12.1 The explicit costs of a rm are the wages it must pay to hire labor, the interest to borrow money capital, and the rent on land and buildings used

A connected component of a graph is a Figure 15.5 A disconnected graph subgraph that is maximally connected, that is, a connected subgraph with the property that any larger subgraph that contains it is disconnected. The graph in Example 15.5 has five connected components, of sizes 3, 1, 4, 2, and 2. Theorem 15.3 Every graph is a union of a unique set of connected components. A walk is closed if its two end points are the same vertex. A cycle is a closed walk of length at least 3 whose interior vertices are all distinct. EXAMPLE 15.6 Graph Cycles