The Sum Numbers and the Integral Sum Numbers of $\overline{P_n}$ and $\overline{F_n}$

Abstract

Let $G=(V,E)$ be a simple graph with the vertex set $V$ and the edge set $E$. $G$ is a sum graph if there exists a labelling $f$ of the vertices of $G$ into distinct positive integers such that $uv\in E$ if and only if $f(w)=f(u)+f(v)$ for some vertex $w\in V$. Such a labelling $f$ is called a sum labelling of $G$. The sum number $\sigma (G)$ of $G$ is the smallest number of isolated vertices which result in a sum graph when added to $G$. Similarly, the integral sum graph and the integral sum number $\zeta (G)$ are also defined. The difference is that the labels may be any distinct integers.
In this paper, we will determine that
$$\{\begin{array}{l}0=\zeta (\overline{P_4})<\sigma (\overline{P_4})=1;\\ 1=\zeta (\overline{P_5})<\sigma (\overline{P_5})=2;\\ 3=\zeta (\overline{P_6})<\sigma (\overline{P_6})=4;\\ \zeta (\overline{P_n})=\sigma (\overline{P_n})=0,\text{ for }n=1,2,3;\\ \zeta (\overline{P_n})=\sigma (\overline{P_n})=2n–7,\text{ for }n\ge 7;\end{array}$$ and $$\{\begin{array}{l}0=\zeta (\overline{F_5})<\sigma (\overline{F_5})=1;\\ 2=\zeta (\overline{F_5})<\sigma (\overline{F_6})=2;\\ \zeta (\overline{F_c})=\sigma (\overline{F_n})=0,\text{ for }n=3,4;\\ \zeta (\overline{F_n})=\sigma (\overline{F_n})=2n–8,\text{ for }n\ge 7.\end{array}$$