A Characterization of Triangle-Free Cyclic Graphs With Self-Loops Of Rank 3

Johnny Lim

paper

A Characterization of Triangle-Free Cyclic Graphs With Self-Loops Of Rank 3

arXiv:2509.20158

Abstract

Let $G_{S}$ be a self-loop graph as the graph obtained by attaching a self-loop at every vertex in $S \subseteq V (G)$ of a simple graph $G .$ If $G = C_{n}$ is the cycle graphs of order $n$ and $S \neq = \emptyset,$ we show that there are no rank 3 self-loop graphs $(C_{n})_{S}$ for $n \geq 5.$ As a consequence, we determine and construct all possible rank 3 triangle-free self-loop cyclic graph of order at least 4 from $(C_{4})_{S}$ via graph join operations. This provides a partial solution to the characterization problem of rank 3 self-loop graphs.

11 pages, 34 figures. Comments are welcome