Lucas Mean Labeling for Some Graphs

Abstract

A function $f$ is called Lucas mean labeling of graph $G$ if $f:V\left(G\right)\to\{0,1,2,\dots ,L_{p+q}\}$ is injective and the induced function $f:E\left(G\right)\to\{L_1,L_2,\dots ,L_q\}$ defined as
\[
f^*\left(e=uv\right)=\left\{\begin{array}{ll}
\frac{f\left(u\right)+f(v)}{2}& \hbox{if $f\left(u\right)+f\left(v\right)$ is even} \\
\frac{f\left(u\right)+f\left(v\right)+1}{2}& \hbox{if $f\left(u\right)+f\left(v\right)$ is odd}\end{array}
\right.
\]
is bijective. A graph which admits Lucas mean labeling is called Lucas mean graph. In this paper, we proved for some graphs such as path $P_n$, Twigs $T_n$, $P_n\odot K_1$, the graph obtained by the subdivision of the edges of the path $P_n$ in comb $P_n\odot K_1$, $(C_3\odot K_{1,n})$, $\left\langle C_3, K_{1,n} \right\rangle $, $\left\langle C_3^*, K_{1,n} \right\rangle $, $(3,n)$ kite graph, $B_{m,n}$, $K_{1,n,n}\cup K_{1,m,m} $ are Lucas mean graph.