Year: 2025
Type: article
Abstract: Let $$G_S$$ G S be a graph with loops attached at each vertex in $$S \subseteq V(G).$$ S ⊆ V ( G ) . In this article, we develop exact formulae for the number of closed 3- and 4-walks on $$G_S$$ G S in terms of vertex degrees and certain elementary subgraphs of $$G_S.$$ 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 $$G_S,$$ G S , which otherwise may be trivial in the loopless case. Moreover, we study a moment-like quantity $$\mathcal {M}_q(G_S)=\sum ^n_{i=1} |\lambda _i(G_S) - \frac{\sigma }{n}|^q,$$ M q ( G S ) = ∑ i = 1 n | λ i ( G S ) - σ n | q , twisted by the spectral moment $$\textsf{M}_1(G_S)$$ M 1 ( G S ) for $$G_S,$$ G S , and show a positivity result. We also establish that the following ratio inequality holds: $$ \frac{\mathcal {M}_{1}}{\mathcal {M}_{0}} \le \frac{\mathcal {M}_{2}}{\mathcal {M}_{1}} \le \frac{\mathcal {M}_{3}}{\mathcal {M}_{2}} \le \frac{\mathcal {M}_{4}}{\mathcal {M}_{3}} \le \cdots \le \frac{\mathcal {M}_{n}}{\mathcal {M}_{n-1}} \le \cdots . $$ M 1 M 0 ≤ M 2 M 1 ≤ M 3
Author: Johnny Lim
Institution: Universiti Sains Malaysia
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Universiti Sains Malaysia
OpenAlex ID: W4414628716