On super edge-magic deficiency of certain Toeplitz graphs

A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$. Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup nK_1,$ has a super edge-magic labeling, if such integer does not exist we define $\mu_s(G)$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.