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

TL;DR

This paper introduces new graph operations and derives their normalized Laplacian spectra based on component spectra, enabling the construction of cospectral graphs and calculating various graph invariants.

Contribution

The paper defines two novel graph operations and determines their normalized Laplacian spectra in terms of component spectra, extending previous spectral results.

Findings

Derived spectra formulas for the new graph operations.

Constructed infinitely many pairs of normalized Laplacian cospectral graphs.

Calculated spanning trees, Kirchhoff index, and Kemeny's constant for the new graphs.

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 $G_1^S\bowtie (G_2^V\cup G_3^E)$ and $G_1^S\diamondsuit(G_2^V\cup G_3^E)$, respectively. Whereafter, the normalized Laplacian spectra of $G_1^S\bowtie (G_2^V\cup G_3^E)$ and $G_1^S\diamondsuit(G_2^V\cup G_3^E)$ 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 $G_1^S\bowtie (G_2^V\cup G_3^E)$ (resp. $G_1^S\diamondsuit(G_2^V\cup G_3^E)$).