On super edge-magic deficiency of certain Toeplitz graphs

A graph $G$ is called edge-magic if there exists a bijective function $\phi:Vinsert ignore into journalissuearticles values(G);\cup Einsert ignore into journalissuearticles values(G);\to\{1, 2,\dots,|Vinsert ignore into journalissuearticles values(G);|+|Einsert ignore into journalissuearticles values(G);|\}$ such that $\phiinsert ignore into journalissuearticles values(x);+\phiinsert ignore into journalissuearticles values(xy);+\phiinsert ignore into journalissuearticles values(y);=cinsert ignore into journalissuearticles values(\phi);$ is a constant for every edge $xy\in Einsert ignore into journalissuearticles values(G);$, called the valence of $\phi$. Moreover, $G$ is said to be super edge-magic if $\phiinsert ignore into journalissuearticles values(Vinsert ignore into journalissuearticles values(G););=\{1,2,\dots,|Vinsert ignore into journalissuearticles values(G);|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_sinsert ignore into journalissuearticles values(G);$, 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 $\mu_sinsert ignore into journalissuearticles values(G);$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.