On Squared Distance Matrix of Complete Multipartite Graphs

Let $G=K_{n_1,n_2,\cdots ,n_t}$ be a complete $t$-partite graph on $n={\sum }_{i=1}^t{n}_i$ vertices. The distance between vertices $i$ and $j$ in $G$, denoted by $d_{ij}$ is defined to be the length of the shortest path between $i$ and $j$. The squared distance matrix $\mathrm{\Delta }(G)$ of $G$ is the $n\times n$ matrix with $(i,j{)}^{th}$ entry equal to $0$ if $i=j$ and equal to $d_{ij}^2$ if $i\mathrm{\ne }j$. We define the squared distance energy $E_{\mathrm{\Delta }}(G)$ of $G$ to be the sum of the absolute values of its eigenvalues. We determine the inertia of $\mathrm{\Delta }(G)$ and compute the squared distance energy $E_{\mathrm{\Delta }}(G)$. More precisely, we prove that if $n_i\ge 2$ for $1\leq i \leq t$,then$ E_{\Delta}(G)=8(n-t)$andif$ h= |{i : n_i=1}|\geq 1$,then$$8(n-t)+2(h-1)\le E_{\mathrm{\Delta }}(G)<8(n-t)+2h\mathrm{.}$$ Furthermore, we show that for a fixed value of $n$ and $t$, both the spectral radius of the squared distance matrix and the squared distance energy of complete $t$-partite graphs on $n$ vertices are maximal for complete split graph $S_{n,t}$ and minimal for Tur{'a}n graph $T_{n,t}$.