Explain this problem with a graph to understand and explain it step by step.
a) Show that if G is a 2-connected graph containing a vertex that is adjacent to at least three vertices of degree 2, then G is not hamiltonian.
b) The subdivision graph S(G) of a graph G is that graph obtained from G by replacing each edge uv of

Is it TRUE or FALSE that ( and why )
In an undirected graph(with no self loops), if every vertex has degree at least n/2, then the graph is fully connected ?
Thanks

The solution below got cut off. Please let me know what is the total solution:
problem:
The number of strongly connected components in a graph G is k. By how much can this number change if we add a new edge?
solution:
If we add an edge to a biconnected graph with k strongly connected components, then the

Using the theorems from Graph and Digraphs 4ed by G. Chartrand and L. Lesniak:
1 - (a) Let G a graph of order n such that deg v greater than or equal to (n-1)/2 for every v element of V(G).
Prove that G is connected.
(b) Examine the sharpness of the bound in (a).
2 - Prove the every graph G has a path of length sigma

Please help answer the following question. Provide step by step calculations along with detailed explanations to explain how each step works.
If all edges of Kn (a complete graph) have been coloured red and blue, how do we show that either the red graph or the blue graph is connected?

Let G be a graph of diameter at least three. Can you find an upper bound on the diameter of the complement of G? Prove your findings!
Let G be a connected graph and sq(G) be a graph which contains all vertices and edges of G and moreover edges joining every pair of vertices that were in G at distance 2. In other words, xy is

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphi

(See attached file for full problem description with proper symbols)
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? Show that, for , the sphere is path connected.
? Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected.
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The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G.
The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of