Using Lagrange's theorem, find the order of all the potential subgroups of the group
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- Given the function f (a, b, c, d) = ∑m(0,2,10,11,12,14,15) + ∑d(1,8,13), how many X don't cares are not needed in the groups in the the K-map of f?Given the directed graph G below, use diagrams to show: 1) Adjacency-list representation of G 2) Adjacency-matrix representation of G(a) Draw an undirected graph G1 represented by the following adjacency matrix: 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 Does G1 contain a Hamiltonian cycle? Explain (b) Determine which pairs of graphs G1 in part (a) and G2 and G3 shown below are isomorphic and which are not, and explain your answer. If you use any invariant property in your argument, explain why this property is an invariant. Tex our answer. t 2 G2 G3 5. 7 8.
- Find a graph with this sequence of degrees (987753210) or prove that it does not exist.Let G be a group and H be a subgroup of G.(a) Show that H is a normal subgroup of G if and only if Hx = xH for all x ∈ G.(b) Show that if |G : H| = 2, then H is a normal subgroup of G.Show that the graphs contains a Hamiltonian cycle
- ---DON'T CARES IN KARNAUGH MAPS (K-MAPS). 5.) Given this truth table, find the unoptimized formula this corresponds to, along with the version which was optimized with a K-Map. A B C D | O 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 X 0 1 x 0 x 0 x 0 1 0 1 0 X 0 Unoptimized Sum of Products equation 5a) 0 = Draw the K-map and simplify Optimized Sum of Products equation 5b) 0 =Can you prove or provide a counter example for:f a function f has a unique minimal sum-of-products expression then it also has a unique minimalproduct-of-sums expressionLet T:V→V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here Vis the vector space of all formal real linear combinations of the vertices v1,..,v10 of the Petersen graph. Question: compute the spectrum of the adjacency operator Tof the Petersen graph. What do you observe?