Question

3. Find a graph with the given set of properties or explain why no such graph can exist. The graphs do not need to be trees unless explicitly stated. (a) tree, 7 vertices, total degree = 12. (b) connected, no multi-edges, 5 vertices, 11 edges. (c) tree, all vertices have degree <3, 6 leaves, 4 internal vertices. (d) connected, five vertices, all vertices have degree 3.

Answer 1

i explain all the part clearly and draw the graph if possible.Hope you understand.Thank you
w Page - (a) since the thee has a vertices. Then it musut have (7-1)= 6 edges, consequently the so Sum of the degrace of the veradices must be 6x2 =12 So The Aree exist. Theme fore the graph of the thee is 3 i) connected, no - multiple edges, 5 veradices, 11 edges. And since the graph is connected and no multiple edges. So to draw a graph with a vertices There at most 56 = 10 edges ( since there is no multiple edges, but it is given there are 11 edges so we say that this graph does not exist. type of n

pag- (C) thee, alle vendices of degree 23, 6 leaves 4 indesanal venlices And since there are 6 leaves and 4 internal ventices. So the othee has 10 ventices therefore there are (10-1)=g edges. So the sum of the degrees of the vertices is 982=18. Now the in this graph there are 6 deavos 50 6 veralices have degree 1. so the totalt som of the degree of 4 intennial vertices ane (18-621) = 12 - go each internal vertices have degree may be take as 3. Hence this type of free will exist And the graph of the tree is

Page ③ (d) connected, five vestices, All vestices have degree 3. And since, we know that the sum of the da degrees of all the veratices of a Graph is an even in degera Here 5 venticos Here in the graph 5 vendices and All venfices of degree 3. 50 the sum of the degrees of the vertices of the graph is 5x3=15 which is not an even number. So, this type of graph does not exist