The irregularity strength and cost of the union of cliques (Q1916108)

A network \(G(w)\) consists of a graph \(G\) together with a mapping \(w\) of its edge set into the set of positive integers; the value \(w(e)\) is called the weight of the edge \(e\). The strength \(s(G(w))\) of the network \(G(w)\) is the maximum of \(w(e)\) over all edges \(e\) of \(G\). The weighted degree of a vertex \(x\) of \(G\) is the sum of weights of edges which are incident to \(x\). If no two vertices have the same weighted degree, the network \(G(w)\) is called irregular. The irregularity strength \(s(G)\) of \(G\) is the minimum of \(s(G(w))\) over all irregular networks \(G(w)\) with the underlying graph \(G\). A similar concept is the irregularity cost. The main result of the paper is the theorem determining the irregularity strength of the disjoint union \(\bigcup^n_{i= 1} t_i K_{p_i}\) of cliques. Some results concerning the irregularity cost are added.