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
\[\begin{cases}
0 = \zeta(\overline{P_4}) < \sigma(\overline{P_4}) = 1;\\ 1 = \zeta(\overline{P_5}) < \sigma(\overline{P_5}) = 2;\\ 3 = \zeta(\overline{P_6}) < \sigma(\overline{P_6}) = 4;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 0, \text{ for } n = 1, 2, 3;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 2n – 7, \text{ for } n \geq 7; \end{cases}\] and \[\begin{cases} 0 = \zeta(\overline{F_5}) < \sigma(\overline{F_5}) = 1;\\ 2 = \zeta(\overline{F_5}) < \sigma(\overline{F_6}) = 2;\\ \zeta(\overline{F_c}) = \sigma(\overline{F_n}) = 0, \text{ for } n =3,4;\\ \zeta(\overline{F_n}) = \sigma(\overline{F_n}) = 2n – 8, \text{ for } n \geq 7. \end{cases}\]