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

Document Type : Original paper

Authors

1 Faculty of Electronic Engineering, Nis, Serbia

2 Faculty of Electronic Engineering

3 Yenikent Kardelen Konutlari, Selcuklu

Abstract

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 $d_1geq
d_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ 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 $%
K^{+}(G)=sum_{i=1}^nfrac{1}{gamma _{i}^{+}}$ are considered.

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Main Subjects