Fisica

1) Usando el algoritmo euclidiano, halle el mcd de:
a) 40 y 15
b) 80 y 50
c) 35 y 45

9.1 Find all graphs with 2,3 and 4 nodes.

9.2 (a) Is there a graph on 6 nodeswith degrees 2,3,3,3,3,3?
(b) Is there a graph on 6 nodes with degrees 0,1,2,3,4,5?
(c) How many graphs are there on 4 nodes with degrees 1,1,2,2?
(d) How many graphs arethere on 10 nodes with degrees 1,1,1,1,1,1,1,1,1,1?
9.14 Let G be a graph, and let H1 = (V1,E1) and H2 = (V2,E2) be two subgraphs of G
that are connected. Assume that H1 and H2have at least one node in common. Form
their union, i.e., the subgraph H = (V ′,E′), where V ′ = V1 ∪ V2 and E′ = E1 ∪ E2.
Prove that H is connected.
9.15 Determine theconnected components of the graphs constructed in exercise 9.1.

9.16 Prove that no edge of G can connect nodes in different connected components.

9.17 Prove that a node v isa node of the connected component of G containing node
u if and only if g contains a path connecting u to v.

10.2 Prove that connecting two nodes u and v in a graph G by anew edge creates a
new cycle if and only if u and v are in the same connected component of G.

10.3 Prove that in a tree, every two nodes can be connected by a unique path.Conversely,
prove that if a graph G has the property that every two nodes can be connected
by a path, and there is only one connecting path for each pair, then the graph is atree.
10.7 Find all unlabelled trees on 2,3,4 and 5 nodes. How many labelled trees do you
get from each? Use this to find the number of labelled trees on 2,3,4 and 5 nodes.10.9 Consider the following “codes”: (0, 1, 2, 3, 4, 5, 6, 7); (7, 6, 5, 4, 3, 2, 1, 0);
(0,0,0,0,0,0,0, 0); (2,3,1,2,3,1,2, 3). Which o
f these are “father codes” of trees?