Graph Theory- vertex degree

Question

Consider an undirected graph with n vertices, vertex 1 has degree 1, while each vertex 2,3......, n – 1 has degree 4. The degree of vertex n is unknown. Which of the following statement must be TRUE?

a. Vertex n has degree 1.

b. Graph is connected.

c. There is a path from vertex 1 to vertex n.

d. Spanning tree will include the edge connecting vertex 1 and n.

Answer 1

Option a) false.
By handshaking lemma you can argue that degree of n must be odd but it doesn't mean it will be 1 always.
For example 1,4,4,4,4,3 is a valid degree sequence.
B) false.
Suppose the graph has 2 component as degree sequence G1 = 1,4,4,4,4,3 and G2= 4,4,4,4,4.
Which means it is not always true that graph is connected.
C) True
Vertex 1 and vertex n should be a vertex of same component of graph otherwise it will not follow handshaking lemma. Graph is undirected and vertex 1 and n are connected so, there must be a path from 1 to n.
D) false
Not every spanning tree will contain the edge joining 1 to n.

Answer 2

The vertex n....is the vertex with odd degree.....as sum of degree of all the vertex has to be even.......also theorem says that if there are only 2 vertex of odd degree then there is a path connecting those 2 vertices.....hence ans is C.