Decomposing \(\lambda K_{v}\), into the Graphs with Seven Vertices, Seven Edges and one 5-Cycle for any Index \(\lambda\)

Let \(\lambda K_{v}\) be the complete multigraph of order \(v\) and index \(\lambda\), where any two distinct vertices \(x\) and \(y\) are joined exactly by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design of \(\lambda K_{v}\), denoted by \((v, G, \lambda)\)-GD, is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\) and \(\mathcal{B}\) is a collection of subgraphs of \(K_{v}\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_{v}\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). There are four graphs with seven vertices, seven edges, and one 5-cycle, denoted by \(G_i\), \(i=1,2,3,4\). In \cite{9}, we have solved the existence problems of \((v, G_i, 1)\)-GD. In this paper, we obtain the existence spectrum of \((v, G_i, \lambda)\)-GD for any index \(\lambda\).