Summary: If there are integers \(k\) and \(\lambda\neq 0\) such that a total labeling \(f\) of a connected graph \(G= (V,E)\) from \(V\cup E\) to \(\{1,2,\dots,|V|+|E|\}\) satisfies \(f(x)\neq f(y)\) for distinct \(x,y\in V\cup E\) and \(f(u)+ f(v)= k+\lambda f(uv)\) for each edge \(uv\in E\), then \(f\) is called a \((k,\lambda)\)-magically total labeling (\((k,\lambda)\)-mtl for short) of \(G\). Several properties of \((k,\lambda)\)-mtls of graphs are shown.
Necessary and sufficient connections between \((k,\lambda)\)-mtls and several known labelings (such as graceful, odd-graceful, felicitous and \((b,d)\)-edge antimagic total labelings) are given. Furthermore, every tree is proven to be a subgraph of a tree having super \((k,\lambda)\)-mtls.