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