# Clique number of a graph given its order and average degree

Let $$G$$ be simple graph of order $$N$$, and let $$\bar{d}$$ be its average degree.

1. Find the maximum value of $$\omega(G)$$ (the clique number of $$G$$) as a function of $$N$$ and $$\bar{d}$$.

2. Find the minimum value of $$\omega(G)$$ as a function of $$n$$ and $$\bar{d}$$.

3. What are the minimum and maximum values of $$\omega(G)$$ when $$G$$ is $$d$$-regular?

• This seems like homework. What have you tried? Where are you stuck? – Pål GD Feb 18 '19 at 8:53
• I feel like the first part should be a quadratic function(probably). Something like k(k-1)/2 where k is size of maximum clique. I'm not sure how to approach this question when the graph is d-regular though! – Ritwik Feb 18 '19 at 10:13
• When the graph is $d$-regular, certainly $\omega(G) \leq d+1$, and this is achievable (for example, a clique $K_{d+1}$). Conversely, if $d \leq n$, then you can take the graph on $x_1,\ldots,x_n,y_1,\ldots,y_n$ (so $N = 2n$) containing the edges $(x_i,y_{i+j \bmod n})$ for $1 \leq j \leq d$, which is $d$-regular but triangle-free. – Yuval Filmus Feb 18 '19 at 11:15
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