The proposed method is superior to the existed edge-map based method from the experimental comparison.
It then follows that there must exist edges such that, which contradicts our assumption that is a -valid matching.
(3) In a subnet, there exist edges, i.e., sensor sensor, sensor actor, actor actor and actor gateway.
(2) For every subnet in the network topology, there exist edges which describe communication namely, sensor sensor, actor actor, gateway gateway, sensor-actor, sensor-gateway and actor-gateway.
Since X F and Y F are cliques in GΔ, there exist edges { x1, x2} and { y1, y2}.
Since H1 and H2 are full components, there exist edges { e, e1} and { e, e2} in LG (P ) where e1∈ V(H1) and e2∈ V(H2).
This is possible because there exist edges from every clique member to v i and v j, and those edges are compatible with the rest of the match's edges - again true by construction of H.
The map leaves an edge fixed if there exists an edge which coincides with the edge.
Hence, we can conclude that for every edge f∈ Q′, there exists an edge e′∈(Q∖{ e∗}), where Ψ′(f)= Ψ(e′).
The existed zigzag edges and a special potential configuration in a single GNR make it promise for fabrication of graphene based devices as triode and spintronic device.
