Determine whether the graph is Hamiltonian 
how to check this is graph is Hamiltonian or not can u any help me  
 A: Hint.  If the graph is Hamiltonian it has a subgraph $C_{11}$ (a cycle of length $11$).  By considering the cycle you can prove the following:

given any six vertices of the graph, at least two of them must be adjacent.

But in the graph you can find, by trial and error, six vertices with no two adjacent.

Edit.  Even easier: the graph is bipartite, with $6$ vertices "on one side" and $5$ on "the other".
A: Hamiltonian graph problems can be very hard. In this instance, the small size of the graph, its symmetry, and its many vertices of degree $3$ make it easy. We start by assuming the existence of a Hamiltonian cycle $C$; we will either find the cycle or arrive at a contradiction showing that none exists; at some point we might have to split the problem into cases, but let's hope not.


*

*At least one of the edges $di,bk$ must be in $C$. (Why?) By symmetry, we may assume that $di$ is in $C.$

*Either $ad$ or $cd$ is in $C$. By symmetry, we may assume that $ad$ is in $C.$

*The edge $ej$ must be in $C.$ (Why?)

*Either $ae$ or $ei$ is in $C.$ By symmetry, we may assume that $ae$ is in $C.$

*The edges $bc,bk$ are in $C.$
You can take it from here.
