Topogenic graphs
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Given any simple graph G = (V,E) , not necessarily finite, and a ‘ground set’ X , a set-indexer of G is an injective set-valued function f : V → 2^X such that the induced edge function f^⊕ : E(G) → 2^X − {Ø} , defined by f^⊕(uv) = (f(u) − f(v)) ∪ (f(v) − f(u)) = f(u) ⊕ f(v) for each uv ∈ E , is also injective. A set-indexer f of G is topogenic if the family f(V ) ∪ f^⊕(E)is a topology on X , where f(V ) = {f(u) : u ∈ V } and f^⊕(E) = { f^⊕(e) :e ∈ E} . The main aim of this paper is to study topogenic graphs which are defined as those which admit topogenic set-indexers.
Abstract
Given any simple graph G = (V,E) , not necessarily finite, and a ‘ground set’ X , a set-indexer of G is an injective set-valued function f : V → 2^X such that the induced edge function f^⊕ : E(G) → 2^X − {Ø} , defined by f^⊕(uv) = (f(u) − f(v)) ∪ (f(v) − f(u)) = f(u) ⊕ f(v) for each uv ∈ E , is also injective. A set-indexer f of G is topogenic if the family f(V ) ∪ f^⊕(E)is a topology on X , where f(V ) = {f(u) : u ∈ V } and f^⊕(E) = { f^⊕(e) :e ∈ E} . The main aim of this paper is to study topogenic graphs which are defined as those which admit topogenic set-indexers.
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