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 $\{\pm 1\}$. 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.

16 pages