On Spectrum of Neighbourhood Corona Product of Signed Graphs

Abstract

Given two signed graphs $\Gamma_1$ with nodes $\{u_1,u_2,\cdots,u_n\}$ and $\Gamma_2$, the neighbourhood corona, $\Gamma_1*\Gamma_2$ is the signed graph obtained by taking one copy of $\Gamma_1$ and $n_1$ copies of $\Gamma_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of $\Gamma_2$ by a new signed edge. In this paper we will determine the condition for $\Gamma_1*\Gamma_2$ to be balanced. We also determine the adjacency spectrum of $\Gamma_1*\Gamma_2$ for arbitrary $\Gamma_1$ and $\Gamma_2$, and Laplacian and signless Laplacian spectrum of $\Gamma_1*\Gamma_2$ for regular $\Gamma_1$ and arbitrary $\Gamma_2$, in terms of the corresponding spectrum of $\Gamma_1$ and $\Gamma_2$.