Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs

Document Type : Original paper


1 Faculty of Electronic Engineering, Nis, Serbia

2 Faculty of Electronic Engineering

3 Yenikent Kardelen Konutlari, Selcuklu


Let $G=(V,E)$, $V=\{v_1,v_2,\ldots,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\geq d_2\geq\cdots\geq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{n\times n}$ and ${D}=\mathrm{diag}d_1,d_2,\ldots , d_n)$ be the adjacency and the diagonal degree matrix of $G$, respectively. Denote by ${\mathcal{L}^+}(G)={D}^{-1/2} (D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. The eigenvalues of matrix $\mathcal{L}^{+}(G)$, $2=\gamma _{1}^{+}\geq \gamma_{2}^{+}\geq \cdots \geq \gamma_{n}^{+}\geq 0$, are normalized signless Laplacian eigenvalues of $G$. In this paper some bounds for the sum $K^{+}(G)=\sum_{i=1}^n\frac{1}{\gamma _{i}^{+}}$ are considered.


Main Subjects

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