Spectra of s-neighbourhood corona of two signed graphs

Abstract

A signed graph $S=(G, \sigma)$ is a pair in which $G$ is an underlying graph and $\sigma$ is a function from the edge set to $\{\pm1\}$. For signed graphs $S_{1}$ and $S_{2}$ on $n_{1}$ and $n_{2}$ vertices, respectively, the signed neighbourhood corona $S_{1} \star_s S_{2}$ (in short s-neighbourhood corona) of $S_{1}$ and $S_{2}$ is the signed graph obtained by taking one copy of $S_{1}$ and $n_{1}$ copies of $S_{2}$ and joining every neighbour of the $i$th vertex of $S_{1}$ with the same sign as the sign of incident edge to every vertex in the $i$th copy of $S_{2}$. In this paper, we investigate the adjacency, Laplacian and net Laplacian spectrum of $S_{1} \star_s S_{2}$ in terms of the corresponding spectrum of $ S_{1}$ and $ S_{2}$. We determine $(i)$ the adjacency spectrum of $S_{1} \star_s S_{2}$ for arbitrary $S_{1} $ and net regular $ S_{2}$, $(ii)$ the Laplacian spectrum for regular $S_{1} $ and regular and net regular $ S_{2}$ and $(iii)$ the net Laplacian spectrum for net regular $S_{1} $ and arbitrary $ S_{2}$. As a consequence, we obtain the signed graphs with $4$ and $5$ distinct adjacency, Laplacian and net Laplacian eigenvalues. Finally, we show that the signed neighbourhood corona of two signed graphs is not determined by its adjacency (resp., Laplacian, net Laplacian) spectrum.