the condition for a sequence to be potentially $a_{l‎, ‎m}$‎- graphic

the set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $ns_n$‎. ‎a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices‎, ‎and such a graph $g$ is called a realization of $pi$‎. ‎the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$‎. ‎the complete product split graph on $l‎ + ‎m$ vertices is denoted by $overline{s}_{l‎, ‎m}=k_{l} vee overline{k}_{m}$‎, ‎where $k_{l}$ and $k_{m}$ are complete graphs respectively on $l = sumlimits_{i = 1}^{p}r_{i}$ and $m = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers‎. ‎another split graph is denoted by $s_{l‎, ‎m} = overline{s}_{r_{1}‎, ‎s_{1}} veeoverline{s}_{r_{2}‎, ‎s_{2}} vee cdots vee overline{s}_{r_{p}‎, ‎s_{p}}= (k_{r_{1}} vee overline{k}_{s_{1}})vee (k_{r_{2}} vee overline{k}_{s_{2}})vee cdots vee (k_{r_{p}} vee overline{k}_{s_{p}})$‎. ‎a sequence $pi=(d_{1}‎, ‎d_{2},ldots,d_{n})$ is said to be potentially $s_{l‎, ‎m}$-graphic (respectively $overline{s}_{l‎, ‎m}$)-graphic if there is a realization $g$ of $pi$ containing $s_{l‎, ‎m}$ (respectively $overline{s}_{l‎, ‎m}$) as a subgraph‎. ‎if $pi$ has a realization $g$ containing $s_{l‎, ‎m}$ on those vertices having degrees $d_{1}‎, ‎d_{2},ldots,d_{l+m}$‎, ‎then $pi$ is potentially $a_{l‎, ‎m}$-graphic‎. ‎a non-increasing sequence of non-negative integers $pi = (d_{1}‎, ‎d_{2},ldots,d_{n})$ is potentially $a_{l‎, ‎m}$-graphic if and only if it is potentially $s_{l‎, ‎m}$-graphic‎. ‎in this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially $a_{l‎, ‎m}$-graphic and this result is a generalization of that given by j‎. ‎h‎. ‎yin on split graphs‎.