Asian Research Journal of Mathematics

Linear hypergraph set-indexers (LHSIs) associate a graph with a vertex hypergraph and an induced edge hypergraph through injective set-valuations and symmetric-difference edge labels. This study examines structural properties of graphs under LHSIs, with particular emphasis on conditions under which the associated vertex and induced edge hypergraphs are isomorphic. It is shown that a connected graph admitting such an LHSI must be unicyclic and that the cardinality of each edge in the associated hypergraphs is at most two. A construction is established for connected unicyclic graphs in which every vertex has odd degree. The study further identifies graph classes for which the lower and upper LHSI numbers coincide. In particular, for complete bipartite graphs Km,n with m, n ≥ 4, both parameters equal m + n. Exact upper LHSI numbers are also determined for cycles: I^UL (C₃) = 3, I^UL (C₄) = 6, and I^UL (C_n) = 2n for n ≥ 5. The realizability of finite linear hypergraphs as vertex hypergraphs of LHSIs is also investigated. A necessary and sufficient condition is formulated in terms of a spanning subgraph of the complete graph on the hyperedge set whose symmetric-difference edge labels are distinct and themselves form a linear hypergraph. These results clarify several structural restrictions inherent in LHSI constructions.