We are tasked with determining how many edges the graph $G_{1,6}$ has.

Step 1: Understanding the vertex set and edge conditions

The vertex set of $G_{1,6}$ is:

$\{1, 2, 3, 4, 5, 6\}$

The edges in the graph are defined by pairs of vertices $\{i, j\}$ such that $i \neq j$ and $i + j$ is a prime number.

Step 2: Calculate sums and check for primality

We now check the pairs of vertices and compute their sums to determine which are prime:

• $i = 1$:

  • $1 + 2 = 3$ (prime)
  • $1 + 3 = 4$ (not prime)
  • $1 + 4 = 5$ (prime)
  • $1 + 5 = 6$ (not prime)
  • $1 + 6 = 7$ (prime)

• $i = 2$:

  • $2 + 3 = 5$ (prime)
  • $2 + 4 = 6$ (not prime)
  • $2 + 5 = 7$ (prime)
  • $2 + 6 = 8$ (not prime)

• $i = 3$:

  • $3 + 4 = 7$ (prime)
  • $3 + 5 = 8$ (not prime)
  • $3 + 6 = 9$ (not prime)

• $i = 4$:

  • $4 + 5 = 9$ (not prime)
  • $4 + 6 = 10$ (not prime)

• $i = 5$:

  • $5 + 6 = 11$ (prime)

Step 3: List all valid edges

The valid edges correspond to the sums that resulted in a prime number:

• $\{1, 2\}, \{1, 4\}, \{1, 6\}$
• $\{2, 3\}, \{2, 5\}$
• $\{3, 4\}$
• $\{5, 6\}$

Step 4: Count the edges

There are 7 edges in total.

Thus, the graph $G_{1,6}$ has 7 edges.

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Do you want more details on how the sums were checked for primality, or do you have any questions?

5 Related Questions:

1. How do we efficiently check whether a number is prime?
2. What are the properties of a graph with vertices defined by prime-related conditions?
3. Can we generalize this problem for larger $b$ values?
4. How does this graph compare to a complete graph $K_6$?
5. What is the maximum number of edges in a graph where sums of vertices must be prime?

Tip: For small numbers, checking whether a number is prime can be done by testing divisibility up to the square root of the number.