Some New Results on the Curling Number of Graphs

Let $S=S_1{S}_2{S}_3\dots S_n$ be a finite string. Write $S$ in the form $XYY\dots Y=X{Y}^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest value of this integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. Let the degree sequence of the graph $G$ be written as a string of identity curling subsequences say, $X^{k_1}_1\circ X^{k_2}_2\circ X^{k_3}_3 \ldots \circ X^{k_l}_l$\mathrm{.}Thecompoundcurlingnumberof$G$,denoted$cn^c(G)$ is defined to be, $c{n}^n(G)={\prod }_{i=1}^l{k}_i$. In this paper, we discuss the curling number and compound curling number of certain products of graphs.