On the Sparing Number of Certain Graph Structures

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. A mono-indexed element of a graph is an element of $G$ which has the set-indexing number $1$. The Sparing number of a graph $G$ is the minimum number of mono-indexed edges required for a graph $G$ to admit a weak IASI. In this paper, we introduce the notion of conjoined graphs, entwined graphs and floral graphs and study further about the sparing number of various finite graph operations as extensions to our earlier studies and provide some useful results on these types of graph structures.