On Intersection Numbers, Total Clique Covers and Regular Graphs

Let \(\mathcal{A} = \{A_1, \ldots, A_l\}\) be a partition of \([n]\) and \(\mathcal{F} = \{S_1, \ldots, S_m\}\) be an intersecting family of distinct nonempty subsets of \([n]\) such that \(\mathcal{A}\) and \(\mathcal{F}\) are pairwise intersecting families.Then
\[
|\mathcal{F}| \leq \frac{1}{2} \prod_{i=1}^{l} \left( 2^{|A_i|} – 2 \right) + \sum_{S\subsetneqq[l]} \left(\prod_{i\in S}\left( 2^{|A_i|} – 2 \right)\right).
\]
From this result and some properties of intersection graphs on multifamilies, we determine the intersection numbers of \(3\), \(4\), and \(5\)-regular graphs and some special graphs.