Betti Numbers of Edge Ideals of Grimaldi Graphs and Their Complements

Let \(n\ge 2\) $n\ge 2$ be an integer. The Grimaldi graph G(n) is defined by taking the elements of the set \(\{ 0, \ldots , n-1 \}\) $\{0,\dots ,n-1\}$ as vertices. Two distinct vertices x and y are adjacent in G(n) if and only if \(\gcd (x+y, n) =1\) $gcd(x+y,n)=1$ . In this paper, we examine the Betti numbers of the edge ideals of these graphs and their complements.