The Kirchhoﬀ index Kf and the Laplacian–energy– like invariant LEL are two graph invariants deﬁned in terms of the Laplacian eigenvalues. If $\mu_1 \ge\mu_2 \ge...\mu_{n-1} > \mu_n=0$ are the Laplacian eigenvalues of a connected n-vertex graph, then $ Kf = \sum{i=1}{n-1}\frac{1}{\mu_i}$ and $LEL = um{i=1}{n-1}qrt{\mu_i}$. We examine the conditions under which Kf > LEL. Among other results we show that Kf > LEL holds for all trees, unicyclic, bicyclic, tricyclic, and tetracyclic connected graphs, except for a ﬁnite number of graphs. These exceptional graphs are determined.