Closed Walks Of Low Dimension And Twisted Moments On Self-Loop Graphs

Abstract

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 ${\mathcal{M}}_q(G_S)={\sum }_{i=1}^n|{\lambda }_i(G_S)-\frac{\sigma }{n}{|}^q,$ twisted by the spectral moment ${\mathsf{M}}_1(G_S)$ for $G_S,$ and show a positivity result. We also establish that the following ratio inequality holds: \[ \frac{\mathcal{M}_{1}}{\mathcal{M}_{0}} \leq \frac{\mathcal{M}_{2}}{\mathcal{M}_{1}} \leq \frac{\mathcal{M}_{3}}{\mathcal{M}_{2}} \leq \frac{\mathcal{M}_{4}}{\mathcal{M}_{3}} \leq \cdots \leq \frac{\mathcal{M}_{n}}{\mathcal{M}_{n-1}} \leq \cdots. \] 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.

19 pages, 4 figures. To appear in Bull. Malays. Math. Sci. Soc