snhu Directions: Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit. PROBLEM 1 For parts (a) and (b), indicate if each of the two graphs are equal. Justify your answer. a bc a b b ac 0 C abd e e C d C e (a) d e cd Figure 1: Left: An undirected graph has 5 vertices. The vertices are arranged in the form of an inverted pentagon. From the top left vertex, moving clockwise, the vertices are labeled: a, b, c, d, and e. Undirected edges, line segments, are between the following vertices: a and b; a and c; b and c; c and d; e and d; and e and c. Figure 2: Right: The adjacency list representation of a graph. The list shows all the vertices, a through e, in a column from top to bottom. The adjacent vertices for each vertex in the column are placed in a row to the right of the corresponding vertexs cell in the column. An arrow points from each cell in the column to its corresponding row on the right. Data from the list, as follows: Vertex a is adjacent to vertices b and c. Vertex b is adjacent to vertices a and c. Vertex c is adjacent to vertices a, b, d, and e. Vertex d is adjacent to vertices c and e. Vertex e is adjacent to vertices c and d. From the given information, it appears that the two graphs are indeed equal. The adjacency list in Figure 2 accurately represents the connections between the vertices in the undirected graph in Figure 1. Therefore, we can conclude that the two graphs are equal.
snhu a b e (b) C d 00110 -- 00001 10010 10101 01010 Figure 3: An undirected graph has 5 vertices. The vertices are arranged in the form of an inverted pentagon. Moving clockwise from the top left vertex a, the other vertices are, b, c, d, and e. Undirected edges, line segments, are between the following vertices: a and c; a and d; d and c; and e and b. Yes, the two graphs are equal. The undirected graph on the left and the adjacency matrix on the right represent the same information. In the adjacency matrix, a 1 indicates that there is an edge connecting two vertices, and a 0 means there isnt. For example, in the matrix, row 1 column 3 has a 1, indicating an edge between vertex a and c, which is also shown in the graph. Similarly, all other "1"s in the matrix correspond to the edges in the graph. Therefore, the adjacency matrix accurately represents the undirected graph. Hence, the two graphs are equal.
snhu (c) Prove that the two graphs below are isomorphic. 1 2 3 - 5 Figure 4: Two undirected graphs. Each graph has 6 vertices. The ver- tices in the first graph are