The strong intersecting number of a graph (Q2721703)

The line graph \(L(H)\) of a simple hypergraph \(H= (E_1,E_2,\dots, E_n)\) is a simple graph with vertex set \(\{v_1,v_2,\dots, v_n\}\), such that the vertices \(v_i\) and \(v_j\) are adjacent if and only if \(E_i\cap E_j\neq\varnothing\) \((i\neq j)\). The strong intersecting number \(\Omega(G)\) of a simple graph \(G\) is the minimum order of the simple hypergraphs \(H\) with \(L(H)\cong G\). The paper gives a necessary and sufficient condition for \(\Omega(G)=|E(G)|\) with the minimum degree of its vertices \(\geq 2\), and a sharp upper bound for the strong intersecting number.