Examples the right Vi el a Name two paths from vi to v4 es

the right. Vi el a. Name two paths from vi to v4 es b. Name two simple paths from vi to v4 c. Name two walks from vi to v4

Solution

Q 4 ,

a . Name of Two path :

(1) v₁e₁v₂e₂v₃e₅v₄

₍₂₎ v₁e₁v₂e₃v₃e₅v₄

_{(b) Name of two simple path}

(1) v₁e₁v₂e₂v₃e₅v₄

₍₂₎ v₁e₁v₂e₃v₃e₅v₄

_{(c) name of two walk}

(1) v₁e₁v₂e₂v₃e₅v₄

₍₂₎ v₁e₁v₂e₃v₃e₅v₄

Q. 5 (a) in this graph, no Euler circuit possible due to every vertex of this graph has not even degree.

two vertex B and D has odd degree 3 .

according the euler theorm , each vertex of a graph has even degree then this graph has euler cicuit.

(b) in this graph each vertex has even degree so this graph containing Euler circuit.

one Euler circuit is A B C D E F A

Q. 6 from the (Dirac’s theorem). Consider a graph G = (V, E) with n = |V | 3 vertices. If d(v) n/2 for all v V , then G has a Hamiltonian cycle.

in this graph every vertex has degree 8/2= 4 degree(v)>= 4 where 8 is no of vertex in this graph

so graph has halmiltonian circuit v₁ v₂ v₃ v₀ v₁

_{Q 7. both graph has the following obervations}

_{1. both has same number of vertices 10}

_{2. both has same number of edges 15}

_{3. both has same number of degree sequences (3,3,3,3,3,3,3,3,3.3)}

4 but they are not contaning same cycle vector (c₁,c₂,..............c_i.) where c_{i is the number of cycles of lenth i}

_{so graph are isomorphic to each other .}

_{Q. 8 there are there non isomorphic trees with 5 vertices are possible .}

_1.   *--*--*--*--*

_2.