# Clearly this geodesic is of order 48 evidently the

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distance from u to v. Clearly this geodesic is of order 48. Evidently, the diameter of G is at least 47. You could write this as follows: diam(G) 47. (d) Suppose that G is a graph of order 25 and size 99. From this information, we know that any trail in G can be no longer than what number l? Provide the best upper bound. Sorry about the notation, the lower case L. Trails have to have distinct edges. Hence the best bound with the current information is l = 99. (e) Suppose that G is a graph of order 25 and size 99. From this information, we know that any path in G can be no longer than what number l? Provide the best upper bound. Paths have to have distinct vertices. The longest possible would be with 25 vertices and have length 24.

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TEST1/MAD3305 Page 2 of 4 _________________________________________________________________ 2. (20 pts.) Provide mathematical definitions for each of the following terms. (a) A graph G: A graph G consists of a finite nonempty set V of vertices and a set E of 2-element subsets of V called edges. [For convenience, we frequently write V as V(G) and E as E(G) when we are dealing with more than one graph as a time.] (b) Subgraph: A graph H is called a subgraph of a graph G, written H G, if V(H) V(G) and E(H) E(G). (c) Spanning Subgraph: A subgraph H of a graph G is said to be a spanning subgraph if the vertex set of H is the same as the vertex set of G, that is, V(H) = V(G). (d) Bipartite Graph: A graph G is a bipartite graph if there are nonempty subsets U and W of V(G) with U W = V(G), U W = φ , and each edge of G joins a vertex from U and a vertex from W. [The sets U and W are called partite sets.] (e) Diameter: The greatest distance between any two vertices of a connected graph G is called the diameter of G and is denoted diam(G). So diam(G) = max { d(u,v): u,v
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