The normalized Laplacian spectra of subdivision vertex-edge neighbourhood vertex(edge)-corona for graphs

Fei Wen; You Zhang; Wei Wang

paper

The normalized Laplacian spectra of subdivision vertex-edge neighbourhood vertex(edge)-corona for graphs

arXiv:1806.10133

Abstract

In this paper, we introduce two new graph operations, namely, the subdivision vertex-edge neighbourhood vertex-corona and the subdivision vertex-edge neighbourhood edge-corona on graphs $G_{1}$ , $G_{2}$ and $G_{3}$ , and the resulting graphs are denoted by $G_{1}^{S} ⋈ (G_{2}^{V} \cup G_{3}^{E})$ and $G_{1}^{S} ♢ (G_{2}^{V} \cup G_{3}^{E})$ , respectively. Whereafter, the normalized Laplacian spectra of $G_{1}^{S} ⋈ (G_{2}^{V} \cup G_{3}^{E})$ and $G_{1}^{S} ♢ (G_{2}^{V} \cup G_{3}^{E})$ are respectively determined in terms of the corresponding normalized Laplacian spectra of the connected regular graphs $G_{1}$ , $G_{2}$ and $G_{3}$ , which extend the corresponding results of [A. Das, P. Panigrahi, Linear Multil. Algebra, 2017, 65(5): 962-972]. As applications, these results enable us to construct infinitely many pairs of normalized Laplacian cospectral graphs. Moreover, we also give the number of the spanning trees, the multiplicative degree-Kirchhoff index and Kemeny's constant of $G_{1}^{S} ⋈ (G_{2}^{V} \cup G_{3}^{E})$ (resp. $G_{1}^{S} ♢ (G_{2}^{V} \cup G_{3}^{E})$ ).