The fibre--sum of graphs

The operation of fibre–sum, well known in topology as connected sum, joins two given manifolds together, along identical submanifolds. Analogously, the fibre–sum Z of two labelled connected root graphs H₁ and H₂ is obtained by gluing the two root graphs along a common subgraph G. For z₁ ∈ H₁ and z₂ ∈ H₂, G is identical to the vertex–deleted subgraphs H₁ − z₁ and H₂ − z₂ of H₁ and H₂ respectively. The fibre–sum of H₁ and H₂ may be considered as a perturbation of their 0–1–adjacency matrix. We show that the operation is key to establishing the sufficient conditions for the uniqueness of H₁( ≃ H₂), when constructed from G. Thestructural constraints in the fibre–sum restrict the types of terminal vertices z₁ and z₂ in the fibre–sum, edged fibre–sum and theirassociated root graphs, giving rise to molecular electronic devices with specific conductivity behaviour.