For small bridge strength ε the Grover walk on two graphs connected by one edge pulsates with period O(ε^{-1/2}) depending only on each graph's edge count, achieving near-complete periodic transfer when the counts are equal.
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Szegedy walks for arc search in graphs give success probability independent of marked arc when the graph is arc-transitive, ineffective on paths and cycles but effective on K_{n,n}.
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Pulsation of quantum walk between two arbitrary graphs with weakly connected bridge
For small bridge strength ε the Grover walk on two graphs connected by one edge pulsates with period O(ε^{-1/2}) depending only on each graph's edge count, achieving near-complete periodic transfer when the counts are equal.