2025/09/21 by Johnny Lim, Lim, Johnny
Physics and Astronomy · Mathematics · #Complex Network Analysis Techniques #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2509.17035
Let GS be a graph with loops attached at each vertex in S ⊆ V(G). In this article, we develop exact formulae for the number of closed 3- and 4-walks on GS in terms of vertex degrees and certain elementary subgraphs of GS. 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 GS, which otherwise may be trivial in the loopless case. Moreover, we study a moment-like quantity Mq(GS)=∑ni=1 |λi(GS) - \fracσn|q, twisted by the spectral moment M1(GS) for GS, and show a positivity result. We also establish that the following ratio inequality holds: \fracM1M0 ≤ \fracM2M1 ≤ \fracM3M2 ≤ \fracM4M3 ≤ ⋯ ≤ \fracMnMn-1 ≤ ⋯. As a consequence, we obtain lower bounds for the self-loop graph energy E(GS) in terms of Mi, extending some classical bounds.