Assignment Amaq1 Hypercube Graph Q5: Can You Generalize To Q

Assignment Amaq1 Hypercube Graph Q5 Can You Generalize To Qnq2 T

Assignment (AMA) Q1: Hypercube graph Q5. Can you generalize to Qn? Q2: The Petersen graph? Q3: Two opposite corners are removed from an 8-by-8 checkerboard. Prove that it is impossible to cover the remaining 65 squares with 31 dominoes, such that each domino covers two adjacent squares? Q4: Find all possible isomorphism types of the given kind of simple graph? Q5: Draw a forest having ten vertices, seven edges, and three components? Q6: Find all the cut-vertices and cut-edges in this graph below? Q7: Q8: Determine whether the graphs in the given pair are isomorphic? Q9: Draw a digraph that has the given adjacency matrix? Q10: Cartesian product of two graphs (pseudocode)? Q11: Q12: An 8-vertex, 2-component, simple graph with exactly 10 edges and three cycles? Q13: An 11-vertex, simple, connected graph with exactly 14 edges that contains five edge-disjoint cycles? Q14: Prove or disprove: If a simple graph G has no cut-edge, then every vertex of G has even degree? Q15: Prove that if a graph has exactly two vertices of odd degree, then there must be a path between them? Q16: Show that any nontrivial simple graph contains at least two vertices that are not cut-vertices? Q17: Draw the specified tree(s) or explain why on such a tree(s) can exist? – A 14-vertex binary tree of height 3. Q18: Prove that a directed tree that has more than one vertex with in degree 0 cannot be a rooted tree? Instruction: Read the entire text and then respond to the last 3 questions, at least 100 words per question. Be sure to use APA format when citing your sources. What is the Southwest Evolve "Green" Plane? From the outside, the Evolve Green Plane looks just like any other Southwest Boeing aircraft...

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The last three questions pose intriguing challenges within graph theory and related mathematical concepts, demanding a thorough understanding of the properties of trees and their roots in directed graphs, as well as the characterization of vertices within graphs.

Question 16 requires us to demonstrate that any nontrivial simple graph contains at least two vertices that are not cut-vertices. A cut-vertex, also known as an articulation point, is a vertex whose removal increases the number of connected components in the graph (Diestel, 2017). In a nontrivial graph, having more than one vertex connected to the rest of the graph ensures the existence of at least two vertices that are not cut-vertices. This is because, in such graphs, certain vertices, especially those with degrees greater than one, typically serve as critical connectors. However, among the remaining vertices with degree one or those not critical to the overall connectivity, at least two will not be cut-vertices. Their existence is a fundamental property of connected graphs, especially those with more than one vertex, where the connectivity is maintained even after removing some vertices (Bondy & Murty, 2008). This concept has significant implications in network reliability, where the removal of certain nodes does not necessarily disrupt the entire network.

Question 17 involves drawing or explaining why a 14-vertex binary tree of height 3 can exist. A binary tree is a hierarchical data structure where each node has at most two children. The height of a binary tree is defined as the length of the longest path from the root to a leaf. For a binary tree of height 3, the maximum number of vertices is 2^(h+1) - 1, which leads to 2^4 - 1 = 15 vertices. To construct a binary tree with exactly 14 vertices, one would start with a full binary tree of height 3 and remove one leaf node. Such a configuration still maintains the properties of the binary tree, including the maximum of two children per node and the hierarchical structure. This demonstrates the flexibility within the constraints of binary trees, allowing for the precise number of vertices to be achieved by pruning or truncating parts of a full binary tree (Sedgewick & Wayne, 2011).

Question 18 deals with proving that a directed tree with more than one vertex with an in-degree of zero cannot be a rooted tree. A rooted directed tree (or arborescence) is defined as a directed graph with a unique root vertex, which has an in-degree of zero, and every other vertex has exactly one incoming edge (Bang-Jensen & Gutin, 2008). If there are multiple vertices with in-degree zero, the structure cannot qualify as a rooted tree because it would imply multiple roots, leading to multiple disconnected components or cycles, contradicting the fundamental properties of rooted trees. The root is singular by definition; it acts as the unique origin point from which all other vertices are accessible via directed paths. Multiple in-degree-zero vertices would create independent subtrees or disjoint structures, violating the definition of a connected rooted tree (Leighton, 2014). Thus, only one vertex should have in-degree zero for the structure to be a valid rooted tree.

References

• Bang-Jensen, J., & Gutin, G. (2008). Digraphs: Theory, algorithms, and applications. Springer Science & Business Media.
• Bondy, J. A., & Murty, U. S. R. (2008). Graph theory. Springer.
• Diestel, R. (2017). Graph theory. Springer.
• Leighton, F. T. (2014). Introduction to parallel algorithms and architectures: Arrays, trees, hypercubes. Morgan Kaufmann.
• Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed.). Addison-Wesley.