A CHARACTERIZATION OF TRIANGLE-FREE CYCLIC GRAPHS WITH SELF-LOOPS OF RANK 3

A CHARACTERIZATION OF TRIANGLE-FREE CYCLIC GRAPHS WITH SELF-LOOPS OF RANK 3

Johnny Lim

[PDF]

Abstract

Let GS be a self-loop graph as the graph obtained by attaching a self-loop at every vertex in S ⊆ V (G) of a simple graph G. If G = Cn is the cycle graphs of order n and S ̸= ∅, we show that there are no rank 3 self-loop graphs (Cn)S for n ≥ 5. As a consequence, we determine and construct all possible rank 3 triangle-free self-loop cyclic graph of order at least 4 from (C4)S via graph join operations. This provides a partial solution to the characterization problem of rank 3 self-loop graphs.

Keywords

Graphs with self-loops, triangle-free, characterization, rank.