An analytic odd mean labeling of some new results of graphs

Abstract

Let $H=(V,E)$ be a graph with $r$ vertices and $s$ edges. A graph $G$ is said to be an analytic odd mean labeling if there exist a bijective function $h:V(H)\rightarrow \{ 0,1,3,5,...,2s-1\} $ with an induce edge labeling $h^*:E(H) \rightarrow N$ such that for each edge $xy$ with $h(x)<h(y)$,
\[
h^*(xy)=
\begin{cases}
\lceil {\frac{h(y)^2-(h(x)+1)^2}{2}}\rceil ; &\text{if $h(x)\neq 0$}\\
\lceil {\frac{h(y)^2} {2}} \rceil ; &\text{if $h(x)=0 $}\\
\end{cases}
\]
A graph that admits an analytic odd mean labeling is called an analytic odd mean graph. In this paper, we prove that triangular book $B(3,c)$, double triangular book $DB(3,c)$, triangular snake $S_{3,c}$, double triangular snake $D(T_{c})$, butterfly $BF(c,d)$, drum $D_{c}, c \geq 3$, $C_{c} \bigodot P_{c}$, $F_{c} \bigodot K_{1,d}, 1 \leq d \leq 2c-1$, $W_{c} \bigodot K_{1,d}, 1\leq d \leq 2c$, $P_{c}^2 \bigodot K_{1,d}, c \geq 3$ and $1\leq d \leq 2c-3$ are analytic odd mean graphs.