Subgraph summability number of paths and cycles

A vertex labeling of a graph G is a mapping α: V (G) → N, assigning a positive integer value to each vertex. With this we can consider labels of connected induced subgraphs G[U] for U ⊆ V (G), and define α(G[U]) = ∑_u∈U α(u). The subgraph summability number of a connected graph G is the largest integer σ(G) so that label sums of connected induced subgraphs cover the integers 1 through σ(G) for some vertex labeling of G. We investigate subgraph summability labelings for paths and cycles and provide upper and lower bounds.