On Spectrum of Neighbourhood Corona Product of Signed Graphs

Abstract

Given two signed graphs $Γ_1$ with nodes $\{u_1,u_2,\cdots,u_n\}$ and $Γ_2$, the neighbourhood corona, $Γ_1*Γ_2$ is the signed graph obtained by taking one copy of $Γ_1$ and $n_1$ copies of $Γ_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of $Γ_2$ by a new signed edge. In this paper we will determine the condition for $Γ_1*Γ_2$ to be balanced. We also determine the adjacency spectrum of $Γ_1*Γ_2$ for arbitrary $Γ_1$ and $Γ_2$, and Laplacian and signless Laplacian spectrum of $Γ_1*Γ_2$ for regular $Γ_1$ and arbitrary $Γ_2$, in terms of the corresponding spectrum of $Γ_1$ and $Γ_2$.