Let \(G_S\) be a graph with loops attached at each vertex in \(S \subseteq V(G).\) In this article, we develop exact formulae for the number of closed 3- and 4-walks on \(G_S\) in terms of vertex degrees and certain elementary subgraphs of \(G_S.\) We then derive the specific closed walks formulae for several graph families such as complete bipartite self-loop graphs, complete graphs, cycle graphs, etc. We demonstrate that such invariants are non-trivial in \(G_S,\) which otherwise may be trivial in the loopless case. Moreover, we study a moment-like quantity \(\mathcal {M}_q(G_S)=\sum ^n_{i=1} |\lambda _i(G_S) - \frac{\sigma }{n}|^q,\) twisted by the spectral moment \(\textsf{M}_1(G_S)\) for \(G_S,\) and show a positivity result. We also establish that the following ratio inequality holds: \( \frac{\mathcal {M}_{1}}{\mathcal {M}_{0}} \le \frac{\mathcal {M}_{2}}{\mathcal {M}_{1}} \le \frac{\mathcal {M}_{3}}{\mathcal {M}_{2}} \le \frac{\mathcal {M}_{4}}{\mathcal {M}_{3}} \le \cdots \le \frac{\mathcal {M}_{n}}{\mathcal {M}_{n-1}} \le \cdots . \) As a consequence, we obtain lower bounds for the self-loop graph energy \(\mathcal {E}(G_S)\) in terms of \(\mathcal {M}_i,\) extending some classical bounds.
