  Some of the quantum searching models have been given by perturbed quantum
walks. Driving some perturbed quantum walks, we may quickly find one of the
targets with high probability. In this paper, we construct a quantum searching
model finding one of the edges of a given subgraph in a complete graph. How to
construct our model is that we label the arcs by $+1$ or $-1$, and define a
perturbed quantum walk by the sign function on the set of arcs. After that, we
detect one of the edges labeled $-1$ by the induced sign function as fast as
possible. This idea was firstly proposed by Segawa et al. in 2021. They only
addressed the case where the subgraph forms a matching, and obtained by a
combinatorial argument that the time of finding one of the edges of the
subgraph is quadratically faster than a classical searching model. In this
paper, we show that the model is valid for any subgraph, i.e., we obtain by
spectral analysis a quadratic speed-up for finding one of the edges of the
subgraph in a complete graph.
