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Let $D(G)=(d_{ij})_{n\times n}$ denote the distance matrix of a connectedgraph $G$ with order $n$, where $d_{ij}$ is equal to the distance betweenvertices $v_{i}$ and $v_{j}$ in $G$. A graph is called distance integral if alleigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave asufficient and necessary condition for complete $r$-partite graphs$K_{p_{1},p_{2},\ldots,p_{r}}=K_{a_{1}\cdot p_{1},a_{2}\cdotp_{2},\ldots,a_{s}\cdot p_{s}}$ to be distance integral and obtained suchdistance integral graphs with $s=1,2,3,4$. However distance integral completemultipartite graphs $K_{a_{1}\cdot p_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdotp_{s}}$ with $s>4$ have not been found. In this paper, we find and constructsome infinite classes of these distance integral graphs $K_{a_{1}\cdotp_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdot p_{s}}$ with $s=5,6$. The problem ofthe existence of such distance integral graphs $K_{a_{1}\cdot p_{1},a_{2}\cdotp_{2},\ldots,a_{s}\cdot p_{s}}$ with arbitrarily large number $s$ remains open.

Distance integral complete multipartite graphs with $s=5,6$