Given a sequence of undirected connected graphs of vertices with bounded degrees and weights and an additional sequence of killing parameters , one can define a sequence of natural non-normalized measures on non-trivial loops of . Denote by the transition matrix associated with . Assuming that the empirical distributions of the eigenvalues of converge, we determine the limit of the proportion of \{loops which cover every vertex\} as . Let be the Poissonian "loop soup" with intensity . A a corollary, we determine the limit law of the number of the loops covering the whole graph. As two concrete examples, we consider the closed balls in a regular graph and the discrete tori.