Question

What is the maximum possible number of edges in a graph with n vertices if:

(a) the graph is simple?
(b) the graph is acyclic?
(c) the graph is planar?

Try to justify your answers. [Hint: first look at graphs with few vertices.]

Need a clear answer with good neat handwriting please.

Answer 1

(a) A simple graph is an unweighted, undirected graph containing no graph loops or multiple edges. eg. The maximum possible number of edges in a simple graph with n vertices is n(n-1) Justification : - since we have in vertices and any two of them define an edge in a simple graph so no. of maximum possible edges will be niin n-1) (21 . examples - 1. consider this graph with 4 b vertices No. of vertices : 4 No of max possible edges = - 4X14-1) = 4X3 26 By looking at the graph, the no. of max. possible edges is 6. - 2. consider this graph with ó vertices – vertices - 6 No. of max possible edges- 6X16-1) = 5x3 = 15 By looking & at the graph, the no. of max. possible side edges is also 15.

b. An acyclic graph is a graph that has no cycle in it. possible The maximum number of edges in an acyclic graph is M with n vertices of (or nodes here) is n (n-1) Justification - consider any given node say n.. The number of edges it can have to point towards other nodes is n-1: Now let's choose another node na. it can have (n-2) nudes to point to lor edges) since it can't point towards nl other- wise there would be a cycle so we have. max. number edges- (0-1) tin-2) + (n-3) + ---dito (n-1) El example-0-> No of ververtices or nodes - 4 No of maximum possible edges = 14) (4-1)= 3x2=6 looking at the graph, the maximum possible number of edges is 6. vertices - 5 No of max. possible edges = 5x5-1= 5x2 = 10 Looking at the graph, the maximum possible number of edges is lo.

3. be (c). Planar graph: It is a graph that can be drawn in a plane in which the edges only intersect at vertices · The number of maximbom possible edges with n vertices is [3n-6 Justification - het G be a planar graph with an vertices., e edges and f faces . G must be connected By euler's formula Intf-e=2. We assume each face is having three edges as in any planar embedding of G. So, number of edges in G is f. but each edge is shared by exactly two faces, so we have. de = 3f. hence. ntf-e=&2 -) e = n+1-2 Ent (2)me-2 3e= 3n+2e-6 - le= 3n-6)

STORE consider the graph with 4 vertices- No. of maximum possible edges - 314) - 6 - 6. king at the graph, no. of maximum possible edges is also. 6.