  A sequence $S$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical if it has
a realization containing a $K_{p_{1},p_{2},...,p_{t}}$ as a subgraph, where
$K_{p_{1},p_{2},...,p_{t}}$ is a complete t-partite graph with partition sizes
$p_{1},p_{2},...,p_{t} (p_{1}\geq p_{2}\geq ...\geq p_{t} \geq 1)$. Let
$\sigma(K_{p_{1},p_{2},...,p_{t}}, n)$ denote the smallest degree sum such that
every $n$-term graphical sequence $S$ with $\sigma(S)\geq
\sigma(K_{p_{1},p_{2},...,p_{t}}, n)$ is potentially
$K_{p_{1},p_{2},...,p_{t}}$ graphical. In this paper, we prove that $\sigma
(K_{p_{1},p_{2},...,p_{t}}, n)\geq
2[((2p_{1}+2p_{2}+...+2p_{t}-p_{1}-p_{2}-...-p_{i}-2)n
-(p_{1}+p_{2}+...+p_{t}-p_{i})(p_{i}+p_{i+1}+...+p_{t}-1)+2)/2]$ for $n \geq
p_{1}+p_{2}+...+p_{t}, i=2,3,...,t.$
