Keywords: Graph labeling; Edgemagic graphs; Super edgemagic graphs;
In 1998, Enomoto, et al. [3] introduced the concept of super edgemagic graphs. In 2005, Sugeng and Xie, [7] constructed some super edgemagic total graphs. The usage of the word “super” was introduced in [4]. A edgemagic total labeling is a bijection f from $V(G)\cup E\left(G\right)$ to the integers {1,2,…,p+q} with the property that for every edge f(u) + f(v) + f(uv) = s for some constant s, such a labeling is (V )super if f(V(G)) = {1,2,…,p}. A graph G is called edgemagic (resp. super edgemagic) if there exists an edgemagic (resp. super edgemagic) labeling of G.
Recently, Marimuthu and Balakrishnan, [5], introduced the notion of super edgemagic graceful graphs to solve some kind of network problems. A (p, q) graph G with p vertices and q edges is edge magic graceful if there exists a bijection f : $V(G)\cup E\left(G\right)\to \left\{1,2,\dots ,p+q\right\}$ such that $\leftf\left(u\right)+f\left(v\right)f\left(uv\right)\right\text{}=\text{}k$ , a constant for any edge uv of G. G is said to be super edgemagic graceful if f(V(G)) = {1,2,…,p}.
The double star ${S}_{m,n}$ has two adjacent central vertices x and y. There are m leaves(vertices) ${x}_{1},{x}_{2},\dots ,{x}_{m}$ adjacent to x and n leaves(vertices) ${y}_{1},{y}_{2},\dots ,{y}_{n}$ adjacent to y. A (super) edgemagic total labeling of this graph can be specified by the list $\left(\left\{f\left({x}_{1}\right),f\left({x}_{2}\right),\dots ,f\left({x}_{m}\right)\right\},f\left(x\right),f\left(y\right),\left\{f\left({y}_{1}\right),f\left({y}_{2}\right),\dots ,f\left({y}_{n}\right)\right\}\right).$ One solution for ${S}_{2,2}$ to be edgemagic is ({8,11},2,5, {4,10}) with s = 16. One solution for ${S}_{2,2}$ to be super edgemagic is ({6,3},2,5, {4,1}) with s = 16.
Some existing results found in [6]
i. All Paths have edgemagic total labeling.
ii. A Cycle ${C}_{n}$ is super edgemagic if and only if n is odd.
iii. The Complete bipartite graph ${K}_{m,n}$ is super edgemagic if and only if $n=1$ or $m=1.$
In [6], Marr and Wallis quoted a research problem “For what m and n does ${S}_{m,n}$ have an edge magic labeling?”. In this paper, we prove the following
i. ${S}_{n,n+2}$ is super edgemagic
ii. ${S}_{n,n}$ is super edgemagic if n is even.
Theorem 4.1: The double star ${S}_{n,n+2}$ is super edgemagic.
Proof: Let $G\cong {S}_{n,n+2}$ has two adjacent central vertices x and y with n vertices ${x}_{1},{x}_{2},\dots ,{x}_{n}$ adjacent to $x$ and $n+2$ vertices ${y}_{1},{y}_{2},\dots ,{y}_{n+2}$ adjacent to $y$ . Clearly $p=2n+4$ and $q=2n+3$. Define a total labeling $f:V\left(G\right)\cup E\left(G\right)\to \left\{1,2,\dots ,p+q\right\}$ by $f\left(x\right)=2;\text{}f\left(y\right)=4;\text{}f\left({y}_{i}\right)=2i1$ and $f\left({x}_{j}\right)=2j+4,$ for all $1\le i\le n+2and1\le j\le n.$
Now the edges of G can be labeled as shown in Table 1 and Table 2.
f 
${y}_{1}$ 
${y}_{2}$ 
... 
${y}_{i}$ 
... 
${y}_{n+1}$ 
${y}_{n+2}$ 
y 
$4n+7$ 
$4n+5$ 

$4n\left(2i9\right)$ 

$2n+7$ 
$2n+5$ 
f 
${y}_{1}$ 
${y}_{2}$ 
... 
${y}_{i}$ 
... 
${y}_{n+1}$ 
${y}_{n+2}$ 
x 
$4n+4$ 
$4n+2$ 

$4n\left(2i6\right)$ 

$2n+8$ 
$2n+6$ 
Proof: Let $G\cong {S}_{n,n}$ has two adjacent central vertices x and y with n vertices
${x}_{1},{x}_{2},\dots ,{x}_{n}$ adjacent to $x$ and $n$ vertices ${y}_{1},{y}_{2},\dots ,{y}_{n}$ adjacent to $y$. Clearly $p=2n+4$ and $q=2n+3$. Define a total labeling $f:V\left(G\right)\cup E\left(G\right)\to \left\{1,2,\dots ,p+q\right\}$ by
$f$ 
$y$ 
${y}_{1}$ 
$\left(3n+3\right)+\left(\frac{n}{2}\right)$ 
${y}_{2}$ 
$\left(3n+3\right)+\left(\frac{n}{2}1\right)$ 
. 
. 
${y}_{j}$ 
$\left(3n+3\right)+\left(\frac{n}{2}+1j\right)$ 
. 
. 
${y}_{\frac{n}{2}}$ 
$\left(3n+3\right)+1$ 
${y}_{\frac{n}{2}+1}$ 
$\left(2n+2\right)+\left(\frac{n}{2}\right)$ 
. 
. 
${y}_{k}$ 
$\left(2n+2\right)+\left(n+1k\right)$ 
. 
. 
${y}_{n1}$ 
$\left(2n+2\right)+2$ 
${y}_{n}$ 
$\left(2n+2\right)+1$ 
$f$ 
$x$ 
${x}_{1}$ 
$\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}\right)$ 
${x}_{2}$ 
$\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}1\right)$ 
. 
. 
${x}_{j}$ 
$\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}+1j\right)$ 
. 
. 
${x}_{\frac{n}{2}}$ 
$\left(\frac{7n}{2}+3\right)+1$ 
${x}_{\frac{n}{2}+1}$ 
$\left(\frac{5n}{2}+2\right)+\left(\frac{n}{2}\right)$ 
. 
. 
${x}_{k}$ 
$\left(\frac{5n}{2}+2\right)+\left(n+1k\right)$ 
. 
. 
${x}_{n1}$ 
$\left(\frac{5n}{2}+2\right)+2$ 
${x}_{n}$ 
$\left(\frac{5n}{2}+2\right)+1$ 
 Chartrand G, Lesniak L. Graphs and Digraphs. 3rd Ed. Newyork, London, Washington, D.C: Chapman and Hall, Boca Roton; 1996.
 Chartrand G, Zhang P. Chromatic Graph Theory. Boca Roton: Chapman and Hall/CRC; 2009.
 Emonoto H, Llado AS, Nakamigawa T, Ringel G. Super edgemagic graphs. SUT J Math. 1998;34(2):105109.
 Gallian JA. A dynamic survey of graph labeling. Electron J Combin. 2013;16.
 Marimuthu G, Balakrishnan M. Super edge magic graceful graphs. Inform Sci. 2014;287:140151.
 Marr M, Wallis WD. Magic graphs. Boston, Basel, Berlin: Birkhauser; 2013.
 Sugeng KA, Xie W. Construction of Super edge magic total graphs. Proc 16th AWOCA. 2005:303310.