Question

Assume that the graphs in this problem are simple undirected graphs

A. The minimum possible vertex degree in a connected undirected graph of N vertices is:

B. The maximum possible vertex degree in a connected undirected graph of N vertices is:

C. The minimum possible vertex degree in a connected undirected graph of N vertices with all vertex degree being equal is:

D. The number of edges in a completely connected undirected graph of N vertices is:

E. Minimum possible number of vertices of maximum possible vertex degree in a connected undirected graph of N vertices is:

Answer 1

A -

FOR ANY CONNECTED GRAPH EVERY VERTICE MUST BE CONNECTED TO ATLEAST ONE OTHER VERTICE.

SO THE MINIMUM POSSIBLE VERTEX DEGREE CAN BE 1

THINK IT OF AS A PATH WITH THE CORNER VERTEX ARE CONNECTED TO ONLY ONE ADJACENT VERTEX, SO THEIR DEGREE IS 1

B. -

THE MAXIMUM DEGREE IS POSSIBLE IN A CONNECTED GRAPH THAT IS N-1 WHEN THE EDGE IS CONNECTED TO ALL OTHER VERTEX , MEANING THERE IS AN EDGE GOING TO ALL OTHER N-1 VERTICES .

SO MAXIMUM DEGREE IS N-1

C. -

THE MINIMUM POSSIBLE DEGREE WILL BE WHEN THE ALL VERTICES FORM A RING

JUST IMAGINE THEM AS HOLDING HANDS TO EACH OTHER AND FORMING A BIG CIRCLE JUST.

SO THE MINIMUM POSSIBLE DEGREE WILL BE 2{

HEY THIS IS NOT VALID IN A SPECIAL CASE WHEN THERE ARE ONLY 2 VERTICES THEN THERE THE ANSWER WILL BE 1 }

D. -

FOR ANY CONNECTED GRAPH THERE WILL BE ALWAYS

N COMINATION 2 WAYS

SO NUMBER OF EDGES = N(N-1)/2

E-

JUST IMAGINE A VERTEX WITH ALL THE EDGES GOING OUTWARDS , LIKE HAVE YOU JUST LIKE THE DIAGRAM AS SHOWN SO

ANSWER =

1

lo vr su 3 degree - 2. degree = 2 = 3-1 {N-19 3. res 4. Ei 5. Er 3x23 2

ASK NY DOUBTS IN THE COMMENTS.