Edge bi-magic total labeling on signed necklace

Edge bi-magic total labeling on signed necklace

**Authors : **

R. Chitra, ^{1 *} and S.P. Subbiah ^{2}

**Author Address : **

^{1} Department of Mathematics, The Madura College, Madurai-625011 Tamil Nadu, India.

^{2} P. G. and Research Department of Mathematics, Mannar Thirumalai Naicker College, Madurai-625004 Tamil Nadu, India.

*Corresponding author.

**Abstract : **

Gracefulness and Skolem gracefulness on signed graphs are discussed in {{cite{muktitar} and cite{muktitarsko}}}. No other types of labeling are studied on signed graphs. Here we have introduced a new type of labeling namely, edge bi-magic total labeling on signed graphs and investigates such labeling on some classes of signed graphs.

An injective labeling $f: V cup E^+ cup E^- ightarrow {0,1,2,ldots,p+q}$ on a signed graph $S(V,E, sigma)$ is said to be edge bi-magic total if $f(u)+f(v)+sigma(uv).f(uv)=k_1$ for all $uv in E^+$ and $f(u)+f(v)+sigma(uv).f(uv)=k_2$ for all $uv in E^-,$ where, $k_1 > 0$ and $k_2 > 0$. The positive magic weight of~ $f$ is $w^+=k_1$ and $w^-=k_2$ is the negative magic weight of $f$ on the signed graph $S.$

The positive upper strength $(S^+)$ of~ a signed graph is equal to $max {w_1^+,w_2^+,w_3^+,ldots,w_m^+},$ where $w_1^+,w_2^+,w_3^+,ldots,w_m^+$ are positive magic weights of $f_1, f_2,f_3,ldots,f_m$ on $S$ respectively. The positive lower strength $(s^+)$ of ~a signed graph is equal to $min{w_1^+,w_2^+,w_3^+,ldots,w_m^+}.$

Similarly the negative upper strength $(S^-)$ of~ a signed graph is equal to $max {w_1^-,w_2^-,w_3^-,ldots,w_n^-},$ where $w_1^-,w_2^-,w_3^-,ldots,w_n^-$ are negative magic weights of $f_1, f_2,f_3,ldots,f_n$ on $S$ respectively. The negative lower strength $(s^-)$ of ~a signed graph is equal to $min{w_1^-,w_2^-,w_3^-,ldots,w_n^-}.$ This is total in nature because all the vertices and edges are labeled. All positive and negative magic weights are always greater than or equal to zero, no negative weights are taken in to account.

In this paper, we introduce and investigate the edge bi-magic total labeling on signed necklace.

**Keywords : **

Edge bi-magic total labeling, Positive magic strength, Negative magic strength, Signed Necklace.

**DOI : **

**Article Info : **

*Received : * December 21, 2018; *Accepted : * February 11, 2019.