arXiv subjects
Publications and source records attributed to Eunice Gogo Mphako-Banda.
In this paper, we study some graph theoretical properties of two derivative Euler Phi function set-graphs. For the Euler Phi function $ϕ(n)$, $n\in \mathbb{N}$, the set $S_ϕ(n) =\{i:\gcd(i,n)=1, 1\leq i \leq n\}$ and the vertex set is $\{v_i:i\in S_ϕ(n)\}$. Two graphs $G_d(S_ϕ(n))$ and $G_p(S_ϕ(n))$, defined with respect to divisibility adjacency and relatively prime adjacency conditions, are studied.
The closed neighbourhood $N[v]$ of a vertex $v$ of a graph $G$, consisting of at least one vertex from all colour classes with respect to a proper colouring of $G$, is called a rainbow neighbourhood in $G$. The minimum number of vertices and the maximum number of vertices which yield rainbow neighbourhoods with respect to a chromatic colouring of $G$ are called the minimum and maximum rainbow neighbourhood numbers, denoted by $r^-_χ(G)$, $r^+_χ(G)$ respectively. In this paper, by a colour, we mean a solid colour and by a transparent colour, we mean the fading of a solid colour. The fading numbers of a graph $G$, denoted by $f^-(G)$, $f^+(G)$ respectively, are the maximum number of vertices for which the colour may fade to transparent without a decrease in $r^-_χ(G)$ and $r^+_χ(G)$ respectively.