论文标题
在完整多明图的平方距离矩阵上
On Squared Distance Matrix of Complete Multipartite Graphs
论文作者
论文摘要
令$ g = k_ {n_1，n_2，\ cdots，n_t} $为$ n = \ sum_ {i = 1}^t n_i $ vertices上的完整$ t $ -partite图。 $ d_ {ij} $表示的顶点$ i $和$ j $之间的距离定义为$ i $和$ j $之间最短路径的长度。 $ g $的平方距离矩阵$δ（g）$是$ n \ times n $矩阵，$（i，j）^{th} $输入等于$ 0 $，如果$ i = j $，则等于$ d_ {ij}^2 $，如果$ i \ i \ i \ neq j $。我们将$ g $的平方距离能量$e_δ（g）$作为其特征值的绝对值之和。我们确定$δ（g）$的惯性，并计算平方距离能量$e_Δ（g）$。更准确地说，我们证明，如果$ n_i \ geq 2 $以$ 1 \ leq i \ leq t $，则$e_Δ（g）= 8（n-t）$，如果$ h = | \ {i：n_i = n_i = 1 \} | \ geq 1 $，那么此外，我们表明，对于$ n $和$ t $的固定价值，平方距离矩阵的光谱半径和$ n $ pertices上完整的$ t $ - 粒子图的平方距离能量是完整的拆分图$ s_ {n，t} $的最大值，也是tur {á} n Graph $ t $ t_ n，
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 $Δ(G)$ 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 $i \neq j$. We define the squared distance energy $E_Δ(G)$ of $G$ to be the sum of the absolute values of its eigenvalues. We determine the inertia of $Δ(G)$ and compute the squared distance energy $E_Δ(G)$. More precisely, we prove that if $n_i \geq 2$ for $1\leq i \leq t$, then $ E_Δ(G)=8(n-t)$ and if $ h= |\{i : n_i=1\}|\geq 1$, then $$ 8(n-t)+2(h-1) \leq E_Δ(G) < 8(n-t)+2h.$$ 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{á}n graph $T_{n,t}$.
