ScholarWorks@성균관대학교

초록

For a connected graph Gamma$\Gamma$ with order n, size m and diameter d, the distance signless Laplacian matrix DQ(Gamma)$\mathcal{D}{<}^{>}\mathcal{Q}(\Gamma )$ is defined as DQ(Gamma)=Tr(Gamma)+D(Gamma)$\mathcal{D}{<}^{>}\mathcal{Q}(\Gamma )=\text{Tr}(\Gamma )+\mathcal{D}(\Gamma )$, where Tr(Gamma)$\text{Tr}(\Gamma )$ is the diagonal matrix of vertex transmissions and D(Gamma)$\mathcal{D}(\Gamma )$ is the distance matrix of Gamma$\Gamma$. The eigenvalues of DQ(Gamma)$\mathcal{D}{<}^{>}\mathcal{Q}(\Gamma )$ are the distance signless Laplacian eigenvalues of Gamma$\Gamma$ and are denoted by partial derivative 1 >=partial derivative 2 >=& ctdot;>=partial derivative n${\partial }_1\ge {\partial }_2\ge \cdots \ge {\partial }_n$. The largest eigenvalue partial derivative 1${\partial }_1$ is called the distance signless Laplacian spectral radius. Let Mk(Gamma)=& sum;i=1k partial derivative i$M_k(\Gamma )={\sum }_{i=1}{<}^{>}k{\partial }_i$ and Nk(Gamma)=& sum;i=0k-1 partial derivative n-i$N_k(\Gamma )={\sum }_{i=0}{<}^{>}k-1{\partial }_{n-i}$ be the sum of k-largest and the sum of k-smallest distance signless Laplacian eigenvalues of Gamma$\Gamma$, respectively. In this paper, we obtain the upper bounds for Mk(Gamma)=& sum;i=1k partial derivative i$M_k(\Gamma )={\sum }_{i=1}{<}^{>}k{\partial }_i$ and determine the extremal cases. Also, we obtain the upper bounds for partial derivative 1${\partial }_1$ and determine the extremal graphs. As a consequence, we obtain the lower bounds for Nk(Gamma)=& sum;i=0k-1 partial derivative n-i$N_k(\Gamma )={\sum }_{i=0}{<}^{>}k-1{\partial }_{n-i}$ and for smallest eigenvalue partial derivative n${\partial }_n$ and determine the extremal graphs. Moreover, we obtain the upper bounds for the sum of the squares of the vertex transmissions and sum of the squares of the distances of the vertices and show that the bounds are best possible in each case. As an application, we obtain the upper bounds for the distance signless Laplacian energy of graphs and determine the extremal cases.

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