Linear Hypergraph Set-Indexers of a Graph - m-hikari.com.pdfVIP

International Mathematical Forum, 5, 2010, no. 68, 3359 - 3370 Linear Hypergraph Set-Indexers of a Graph B. D. Acharya Srinivasa Ramanujan Center for Intensification of Interaction in Interdisciplinary Discrete Mathematical Sciences (SRC-IIIDMS) University of Mysore Mysore - 570005, India devadas.acharya@ K. A. Germina Research Center PG Department of Mathematics Mary Matha arts Science College Vemom P.O., Mananthavady - 670645, India srgerminaka@ Viji Paul Department of Mathematics WMO Arts Science College Muttil, Kerala, India vijipaul10@ Abstract Given any graph G = (V, E) and an arbitrary nonempty ‘ground set’ X , a linear hypergraph set-indexer (LHSI) is a set-valued function f : V (G) → 2X assigning nonempty subsets of X to the vertices of G and satisfying the following conditions: (i) the ordered pair Hf (G) = (X, f (V )), where f (V ) = {f (v) : v ∈ V (G)}, is a ‘simple’ hypergraph, in the sense that f is injective and ∪v ∈V (G)f (v) = X, (ii) Hf (G) is a linear hypergraph, in the sense that |f (u) ∩ f (v)| ≤ 1 ∀ u, v ∈ V, (iii) the induced set-valued function f ⊕ : E → 2X defined by f ⊕ (uv) = f (u)⊕f (v),∀ uv ∈ E is injective, and (iv) Hf ⊕ (G) = (X, f ⊕ (E)), where f ⊕ (E) = {f ⊕ (e) : e ∈ E }, is a linear hypergraph. In this paper, we initi