The connected sum Z of two root graphs of order n is obtained by gluing them together along a common subgraph G of order n − 1. The two vertices of Z not in G are called terminal vertices. The edged connected sum Z + e is obtained from Z by adding the edge joining the terminal vertices. We consider the case when the root graphs have the same μ–eigenspace of the 0–1 adjacency matrix of dimension one. We show that the μ–eigenspace imposes structural constraints on Z and Z + e, depending on the type of the two vertices. For μ = 0, we investigate the electrical behaviour of a molecular electronic device with structure Z or Z + e, connected at the terminal vertices in a circuit across a small bias voltage. It transpires that the device will be a conductor or insulator depending on the type of the terminal vertices in the 0–eigenspace. We show that conductivity or its barring distinguishes between Z and Z + e.

Connected sum of graphs as molecular electronic devices

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