Given the following minterm expansion: F(a, b, c, d) = Em(0, 2, 3, 4, 8, 10, 11, 15) Draw the K-Map Find out all essential prime implicants, do it systematically. Do you need additional loops to cover all the minterms. If yes, find the additional p implicants. Get the minimum SOP.

This is a practice problem I need help with.

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.

a. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?

For the 1.square there is a sign change,but it is stated as "no critical point"I couldn't understand it ,please explain me

Please do question 6 (it is in the photo) with python coding.

Long chain of friends: You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k distinct people, such as x1 knows x2, x2 knows x3, and xk-1 knows xk. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)

We want to tile an n × 1 strip with 1 × 1 tiles that are GREEN (G), BLUE (B), and RED (R), 2 × 1PURPLE (P) and 2 × 1 ORANGE (O) tiles. GREEN, BLUE and PURPLE tiles cannot be next to each other, and there should be no two PURPLE or three BLUE or GREEN tiles in a row (for ex., GGOBR is allowed, but GGGOBR,GROPP and PBOBR are not). Give a formula for the number of such tilings. Your solution must include a recurrence equation (with initial conditions!), and a full justification. You do not need to solve it.

I need the answer as soon as possible

We recollect that Kruskal's Algorithm is used to find the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will first sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.