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

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.