The spectrum of neighborhood corona of graphs

Indulal Gopalapillai

Given two graphs $G_1$ with vertices $\{v_1,v_2,\dots,v_n\}$ and $G_2$, the neighbourhood corona, $G_1 \bigstar G_2$ is the graph obtained by taking $n$ copies of $G_2$ and for each $i$, making all vertices in the $i^{th}$ copy of $G_2$ adjacent with the neighbours of $v_i$, $i=1,2,\dots,n$. In this paper a complete description of the spectrum and eigenvectors of $G_1 \bigstar G_2$ is given when $G_2$ is regular, thus adding to the class of graphs whose spectrum is completely known.