The uploaded image contains several mathematical problems, primarily in graph theory, group theory, recurrence relations, and permutation order. Let's address one of the problems in detail. Let me know which specific question you want solved. Here's a summary of the problems:

1. Q6 (Graph Theory):

   • Determine if the given graph has a Hamiltonian circuit (a circuit that visits every vertex exactly once).
   • Construct the circuit if it exists.

2. Q7 (Recurrence Relations and the Five Color Theorem):

   • (a) State and prove the Five Color Theorem.
   • (b) Solve two recurrence relations:
     i. $a_r = 6a_{r-1} - 11a_{r-2} + 6a_{r-3}, a_0 = 1, a_1 = 2, a_2 = 6$
     ii. $a_r - 6a_{r-1} + 9a_{r-2} = r3^r$.

3. Q8 (Group Theory):

   • (a) State and prove Lagrange's theorem, and prove that a group of prime order is cyclic.
   • (b) Check if a set of $3 \times 3$ matrices is a group under matrix multiplication.
   • (c) Find the order of permutations: $(124)(356)$, $(123)(35)$.

4. Q9 (Group Isomorphisms):

   • (a) Determine if the mappings are isomorphisms.
     i. $f: G \to G, f(x) = x^3$, where $G$ is the group of all real numbers under addition.
     ii. $f: G_1 \to G_2, f(x) = 2^x$, where $G_1$ is the group of real numbers under addition, and $G_2$ is the group of all positive real numbers under multiplication.
   • (b) Prove that polynomials with integer coefficients form a commutative ring under addition and multiplication.
   • (c) Find the left cosets of $H = \{0, \pm 3, \pm 6, \pm 9, \dots\}$ in $\mathbb{Z}$ (group of integers under addition).

Please let me know which specific question you'd like solved, or if you'd like me to provide an overview of multiple parts!