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I see that:

For a cycle graph $n\ge3$, $C_3$ has $1$ independent set, $C_4$ has $2$ independent set, $C_5$ s set, $C_6$ $3$ set, $C_7$ $3$ set, $C_8$ $4$ set, $C_9$ $4$ set.

I can't show this trend as an equation or a mathematic expression. Is it possible to show in one?

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up vote 2 down vote accepted

I assume that ‘s’ is a typo for $2$ as the size of a maximal independent set in $C_5$.

If $i(n)$ is the maximal size of an independent set of vertices in $C_n$, you have:

$$\begin{array}{rcc} n:&3&4&5&6&7&8&9\\ i(n):&1&2&2&3&3&4&4 \end{array}$$

You can express this in many ways. The most straightforward is simply to use a two-part definition:

$$i(n)=\begin{cases}\frac{n}2,&\text{if }n\text{ is even}\\\\ \frac{n-1}2,&\text{if }n\text{ is odd}\;.\end{cases}$$

If you know the floor function, also called the greatest integer function, you can simply write


Other ways are more complicated. For instance, you can easily check that


gives the same result as the two-part definition, though I don’t recommend using this form!

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This makes a lot sense than my class notes. I do not know the floor function. Is it common function to use for discrete mathematics? – Uka Nov 7 '12 at 3:24
@Uka: Yes, definitely; so is the ceiling function. This article is a pretty decent introduction to both of them. – Brian M. Scott Nov 7 '12 at 3:26
Thank you for the link and your help! – Uka Nov 7 '12 at 3:31
@Uka: You’re very welcome! – Brian M. Scott Nov 7 '12 at 3:34

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