On Spectrum of Neighbourhood Corona Product of Signed Graphs

Given two signed graphs ${\mathrm{\Gamma }}_1$ with nodes $\{u_1,u_2,\cdots ,u_n\}$ and ${\mathrm{\Gamma }}_2$, the neighbourhood corona, ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$ is the signed graph obtained by taking one copy of ${\mathrm{\Gamma }}_1$ and $n_1$ copies of ${\mathrm{\Gamma }}_2$, and joining every neighbour of the $i^{th}$ node with each nodes of the $i^{th}$ copy of ${\mathrm{\Gamma }}_2$ by a new signed edge. In this paper we will determine the condition for ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$ to be balanced. We also determine the adjacency spectrum of ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$ for arbitrary ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$, and Laplacian and signless Laplacian spectrum of ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$ for regular ${\mathrm{\Gamma }}_1$ and arbitrary ${\mathrm{\Gamma }}_2$, in terms of the corresponding spectrum of ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$.